Math Practice Made Fun: Quizzes, Problems & Answers

The transposition of terms involves passing the terms of an equation from one member to another. In fact, it is a group of numbers that, according to mathematical rules, are allowed to be placed in place of the unknown (or variable) within an equation. The concept of transposing terms is...

Pi is a mathematical value, approximately equal to . This is the commonly used approximation for calculations.Pi is symbolized by .Examples of some mathematical expressions include :...

Adjacent angles are the pair of angles formed when two lines intersect each other. These angles are formed at the point where the intersection occurs, and are adjacent to eachother - hence its name. Another pair of angles that are formed at the intersection of two straight lines are the...

Before going deeper into opposite angles, we will pause a moment to visualize the types of scenarios where this type of angle can be found. To make it easier to understand, we will draw two parallel straight lines cut by a secant or transversal, as shown in the following illustration:...

The collateral angles are a pair of angles that we can find on the same side of a transversal or secant line that intersects two parallel lines, and that are also internal or external with respect to the parallel lines. The sum of the collateral angles equals....

Corresponding anglesDefinition:The corresponding angles are those that are on the same side of the transversal that cuts two parallel lines and are at the same level with respect to the parallel line. The corresponding angles are of the same size.The following image illustrates two pairs of corresponding angles, the first...

Alternate anglesDefinition:Alternate angles are on opposite sides of the transversal that intersects two parallel lines and are not on the same side of the parallel lines to which they belong.Alternate angles are equal.The following sketch illustrates two pairs of alternate angles, one is painted red and the other blue.Identifying Alternate...

What is circumference?This question is not easy to answer and even more complicated to understand. If you imagine any point on a flat surface and a series of points whose distance from that point is identical, then you are looking at a circle.A circumference is the boundary of a circle,...

The base of the power is the number that is multiplied by itself as many times as indicated by the exponent.The base appears as a number or algebraic expression. In its upper right corner, the exponent is shown in small.The base of the power has to stand out clearly since...

Students start learning mathematics as early as elementary school, and as they progress, the subject becomes more and more complicated. Among others, the syllabus devotes a part to geometry and requires students to master different shapes and know how to calculate their area and volume. Are you also studying these...

Rectangular Prisms are made up of different rectangles. When faced with an exercise or exam that asks you to calculate the surface area of a rectangular Prism, use the formula below....

The commutative property of addition lets us change the position of addends (numbers being added together) that are being added together in an expression without changing the end result - no matter how many addends there are!We can use the commutative property in simple expressions as well as algebraic expressions,...

The commutative property is an algebraic principle that allows us to "play" with the position that different elements occupy in multiplication and addition exercises without affecting the final result. Our objective in using the commutative property is to make the resolution of the exercise simpler from the point of view...

Perimeter: calculating the perimeter of a triangleTo calculate the perimeter of a triangle, all you have to do is add its three sides. If you have all the necessary information, you can solve such a problem in a matter of seconds, for example:Formula for the perimeter of a triangle:P =...

The isosceles triangle is a type of triangle that has two sides (legs) of equal length.A consequence of having two sides of equal length implies that also the two angles opposite these sides measure the same....

The equilateral triangle is a triangle that all its sides have the same length.This also implies that all its angles are equal, that is, each angle measures degrees (remember that the sum of the angles of a triangle is degrees and, therefore, these degrees are divided equally...

The triangle is a geometric figure with three sides that form three angles whose sum is always degrees.Its vertices are called and The union between these vertices creates the edges and There are several types of triangles that we will study in this article....

But before explaining what unknowns are, it is important that we review the concept of what a mathematical equation is:An equation is an algebraic expression that includes numbers (fixed values), and also letters with unknown value (unknowns). Our goal is to arrive at a solution to the equation, that is,...

The solution of an equation is, in fact, a numerical value that, if we place it in place of the unknown (or the variable), we will achieve equality between the two members of the equation, that is, we will obtain a "true statement". In first degree equations with one unknown,...

Time units serve the function of quantifying time and have been created in order to organize it productively. They are like a universal language that allows us to measure time.Units of time and conversion that are important to know:In one minute there are seconds.In an hour there are ...

Weight units allow us to quantify weight.Units of weight to know:One gram is equivalent to milligrams.One kilogram is equivalent to grams.One ton is equivalent to kilograms....

Units of measurementOverview:In this article we will learn what units of measurement are, we will know their different types and we will see examples. At the end of the article you will be able to find a table that concentrates all the units of measure.Table of contents:With the units of...

The real line looks like this: a horizontal line in which small equidistant vertical lines are inserted.Real number line ...

Positive, negative numbers and zero are a fundamental topic in algebra, it is very easy to understand it by drawing a number line in which zero is located in the middle.Zero is our reference point.The positive numbers are the same numbers we use to this day and are located to...

For example, if the sides of the rectangle are , its perimeter will be . It is customary to indicate the perimeter by the letter .Important to remember!Rectangles have two pairs of opposite, parallel and equal sides. Therefore, it is enough to know the length of two coincident sides to...

What is the coordinate system?A coordinate system, or more precisely, a Cartesian coordinate system, is a method to precisely present the position of points, whether on a plane (two-dimensional) or in three-dimensional space. In this chapter, we will focus on a coordinate system on a plane, that is, a system...

We can calculate the area of a parallelogram by multiplying one of its sides by its relative height.To understand it better, we can use the following figure and the accompanying formula:It can be seen that: and are the two heights corresponding to the bases and respectively.Area...

The sides of a triangleEvery triangle has three sides. That also works the other way around - if we see a shape with tree sides, it's a triangle.types of triangles based on the sides:The sides allow us to classify the different types of triangles according to their size:Equilateral: All sides...

The sum of the interior angles of a triangle is . If we add the three angles of any triangle we choose, the result will always be . This means that if we know the values of two angles of a triangle we can always calculate, with ease, the value...

Equations are algebraic expressions containing numbers and unknowns. It is important to differentiate these two groups: the numbers are fixed values while the unknowns, as their name indicates, represent unknown values (at least at the beginning), and in most cases we are asked to find out what this value is.For...

A side is the straight line that lies between two points called vertices. An angle is formed between two lines.  A vertex is the point of origin where two or more straight lines meet, thus creating an angle.An angle is created when two lines originate from the same vertex. To clearly...

In the previous articles we studied about the number line and integers. In this article we will explain what opposite numbers are, and how to identify them....

We learned in the previous article about the number line AND we also talked about positive and negative numbers. In this article we move on and call them integers....

the most basic quadratic function:...

A variable is a specific symbol like a Latin letter – // that can change and represent a quantity/value....

Value in mathematics indicates how much something is worth numerically....

In this article, we will learn the three most common ways to solve a quadratic function easily and quickly....

Zero points of a function are its intersection points with the -axis. To find them, we set , we get an equation that can sometimes be solved using a trinomial or the quadratic formula....

Plotting the graph of the quadratic function and examining the roles of the parameters in the function of the form ...

A function is an equation that describes a specific relationship between and . Every time we change , we get a different ....

When we have an exponent on a negative number, we can get a positive result or a negative result. We will know this based on the exponent – whether it is even or odd....

An exponent tells us the amount of times a number is to be multiplied by itself....

The standard representation of the quadratic function looks like this:...

An algebraic fraction is a fraction that contains at least one algebraic expression (with a variable) such as . The expression can be in the numerator or the denominator or both....

1)    Convert the whole number to a fraction 2)    Convert to a multiplication problem, remembering to swap the numerator and denominator of the second fraction 3)    Solve by multiplying fractions...

The topic of reducing and expanding decimal numbers is extremely easy.All you need to remember is the following phrase:...

To convert a decimal fraction to a mixed fraction, we ask ourselves how to read the decimal fraction or in other words, what the last digit represents – if we use the word tenths – we place 10 in the denominator if we use the word hundredths – we place...

Find a common denominator – by expanding and reducing or by multiplying the denominators. (Remember to multiply both the numerator and the denominator)...

There are various types of fractions that need to be known:...

So far we have worked with common two-dimensional figures such as the square or the triangle.Three-dimensional figures are those that extend into the third dimension, meaning that in addition to length and width, they also have height (that is, the figure has depth)....

In this article, we will summarize all the basic rules of mathematics that will accompany you in every exercise - the commutative property of addition, the commutative property of multiplication, the distributive property, and all the others!Shall we begin?...

In this article, we will dive into the world of essential arithmetic rules that are fundamental for tackling a wide variety of mathematical exercises. Mastering these rules will provide you with a solid foundation and allow you to solve problems with greater confidence and precision. From basic operations like addition...

Properties of a regular trapezoid • A quadrilateral with only 2 parallel sides. • Angles resting on the same leg are supplementary to 180 degrees, so the sum of all angles is 360 degrees. • The diagonal of the trapezoid creates equal alternate angles between parallel lines....

Representation using an equation of and ...

Inputing Values into a FunctionIn mathematics, we often assign numerical values to variables in equations or mathematical expressions. A function can be thought of as a machine: you input a value (called the input), the function processes it according to a specific rule, and produces a result (called the output)....

The increasing intervals of a function...

The decreasing intervals of a function...

An acute angle is an angle that measures less than .Acute angles can appear in triangles, parallelograms, and other geometric shapes where there is an angle less than degrees.Acute angleWhich of the 4 angles presented in the figure corresponds to the description of the acute angle?The correct answer is...

Any natural number that is not prime can be factorized as a product of prime numbers. This process is known as breaking down numbers into prime factors.From time to time, to solve a certain exercise in a simpler way, we will have to dfactorize the numbers we are given, and...

When we see any number(positive or negative) raised to a negative power we can convert the expression into a fraction and we will do it as follows: the numerator will be , the denominator will be the base of the power as seen in the original exercise, but now, with...

When we see a number that is not raised to zero, the result will be . Property formula: This property is also concerning algebraic expressions....

In right triangles, we have a condition that already exists in the first place. It refers to the right angle that iss given and that turns a triangle into a right triangle.In the second stage, we will move on to the sides. In every right triangle we have two perpendiculars...

Similar triangles are triangles for which there is a certain similarity ratio, that is, each of the sides of one triangle is in uniform proportion to the corresponding side in the other triangle. In addition, the angles at the same locations are also equal for the two similar triangles....

The similarity ratio is the constant difference between the corresponding sides of the two shapes. That is, if the similarity ratio is , we know that each side of the large triangle is times larger than that of the small triangle....

Similar triangles are triangles whose three angles are equal respectively and also the ratio between each pair of corresponding sides is equal. Two similar triangles are actually larger or smaller versions each other.The ratio of similarity is the ratio between two corresponding sides in two similar triangles.To prove similarities between...

Fractions refer to the number of parts that equal the whole.Suppose we have a cake divided into equal portions, the fraction comes to represent each of the portions into which we have cut the cake. Thus, if we have four equal portions, each of them represents a quarter of the...

A function is a connection between an independent variable and a dependent variable . The relationship between the variables is called a "correspondence rule".An algebraic representation of a function is actually a description of the relationship between the dependent variable and the independent variable by means of...

As we learned in an article on functions, the standard "correspondence rule" is a connection between a dependent variable and an independent variable .By means of a graph or drawing, which gives a visual aspect to the concept of the function. From the graph it is possible to understand...

Function, describes a correlation or coincidence between a dependent variable () and an independent variable (). The legitimacy of this relationship between the variables is called the " correspondence rule "....

The circumference is actually the length of the circular line. It is calculated by multiplying the radius by 2, which has an approximate value of π. It can also be said that the circumference is equal to the the diameter of the circumference multiplied by π (since the diameter is...

A diameter is a section that connects two points that lie on the circumference, that passes through the center of the circle. The diameter is actually twice the radius.As in the case of the radius, as well as in the case of the diameter, there are an infinite number of...

The radius is one of the many elements that exist in a circle. The radius is a segment that connects the center of the circle with any point located on the circle itself. Each circle has an infinite number of radii and their length is exactly the same, that is,...

CircleA circle is a two-dimensional shape where every point on the boundary is equidistant from a central point, called the center. The circle is actually the inner part of the circumference, i.e., the enclosed area inside the circle frame. This distance between the boundary and the center is called radius....

The trapezoid is a quadrilateral defined as having 2 parallel opposite sides. The calculation of the perimeter of the trapezoid is solved using a very simple formula that we will see below: all sides are added together. This type of questions can appear in tests of the first and second...

The plane angle is the one that measures .Plane angleWhich of the 4 angles shown in the figure corresponds to the description of the plane angle?The correct answer is C) 180 °...

This formula is one of the shortcut formulas and it describes the square sum of two numbers.That is, when we encounter two numbers with a plus sign (sum) and they are between parentheses and raised as an expression to the square, we can use this formula. Pay attention - The...

This is one of the shortened multiplication formulas.As can be seen, this formula can be used when there is a multiplication between the sum of two particular elements and the subtraction between the two elements. Instead of presenting them as a multiplication of sum and subtraction, it can be written...

This is one of the shortened multiplication formulas and it describes the square difference of two numbers.That is, when we encounter two numbers with a minus sign between them, that is, the difference and they will be in parentheses and raised as a squared expression, we can use this...

An integral can be defined for all values (that is, for all ). An example of this type of function is the polynomial - which we will study in the coming years.However, there are integrals that are not defined for all values (all ), since if we place certain ...

Rate of change represented by steps in the function graphWe can draw stairs on the graph of the function to see the rate of change. The base of the step will represent the interval in the variables and the height will symbolize the interval in the .The step will...

orwhere : represents the area of the square and –> is the length of the edge (or side) of the square...

Multiplication of Algebraic ExpressionsWe already know that when encountering expressions where a number is added to or subtracted from a variable, we cannot combine them directly. However, when we see an expression where a number multiplies a variable, we can simplify it by applying the multiplication!Multiplying algebraic expressions is the...

Rate of Change of a Function Represented GraphicallyThe rate of change of a function represented graphically allows us to determine in a much more intuitive way whether it is a constant (fixed) or inconstant (not fixed) rate, and also if it is a faster (steeper slope) or slower (more moderate...

When we have a system of equations where one of the equations is linear and the other quadratic we will use the substitution method:We will isolate one variable from an equation, place in the second equation the value of the expression of the variable we have isolated, and in this...

When we have a triangle, we can identify that it is an isosceles if at least one of the following conditions is met: 1) If the triangle has two equal angles - The triangle is isosceles. 2) If in the triangle the height also bisects the angle of the vertex...

Solving an equation using the distributive property is related to the need to open the parentheses as the first step to then be able to simplify similar members. When an equation contains one or more pairs of parentheses, we must start by opening them all and then proceed to the...

Constant Rate of ChangeThe meaning of the constant rate of change of a function can be seen when the variables change in fixed proportions and the do as well. For example, if the constant interval of the is and also that of the is stable and...

The rate of change of a function represented by a table of values allows us to compare the variation of the values of (the independent variable of the function) with the variation of the values of (dependent variable of the function). This comparison enables us to determine if...

Variable Rate of ChangeThe meaning of a variable rate of change of a function can be seen when the variables change in fixed proportions and the change unevenly. A variable rate of change is represented by a line that is not straight, as seen in the following diagram:...

As we have already learned, Coordinate System has two axes and, therefore, any value defined by this coordinate system must include two values. These two values are called "ordered pairs"....

The method to solve an exercise with real numbers, when it involves multiplication and division, is very similar to the one we use when we have to add or subtract real numbers, with the difference that, in this case, we must make use of the multiplication and division table that...

How do we recognize that the quadrilateral in front of us is actually a rectangle? In two quite simple ways!...

Similarity of Geometric FiguresSimilarity in geometry refers to the relationship between two shapes that have the same shape but may differ in size. Two figures are similar if their corresponding angles are equal and the lengths of their corresponding sides are proportional. This means one figure can be obtained by...

The midsegment of a trapezoid divides into two equal parts the two sides from which it extends and, in addition, is parallel to both bases of the trapezoid and measures half the length of these. Let's see the properties of the midsegment of a trapezoid in the following illustration:If ...

Common Factor Extraction Method: Identify the largest free number that we can extract. Then, let's move on to the variables and ask what is the least number of times the appears in any element? Multiply the free number by the variable the same number of times we have found...

The quadratic inequality shows us in which interval the function is positive and in which it is negative - according to the inequality symbol. To solve quadratic inequalities correctly, it is convenient to remember two things:...

Arcs in a CircleAn arc is a portion of the circumference of a circle, the part that is between points on the circle.The arc is part of the circumference of the circle and does not pass inside the circle. Arcs are categorized as either major (larger than half the...

The midsegment of a triangle has three main properties:The midsegment crosses exactly through the middle of the two sides that determine it.The midsegment is parallel to the third side of the triangle.The midsegment measures half the length of the side arranged parallel to it.Let's look at the properties of the...

A chord in a circle is a straight line that connects any points that are on the circle. • The chord passes inside the circle and not over it. • The longest chord in the circle is the diameter • The radius is not a chord....

First step - We will make the decimal point in the dividend (the number we want to divide) disappear by moving it to the right the necessary number of places until it is completely "gone".Second step - In the divisor (the second number in the operation, that is, the number...

The distance from the chord to the center of the circle is defined as the length of the perpendicular from the center of the circle to the chord. Theorems on the distance from the center of the circle:Chords that are equal to each other are equidistant from the center of...

The perpendicular to the chord comes out of the center of the circle, intersecting the chord, the central angle in front of the chord and the arc in front of the chord. Moreover, if there is a section that comes out from the center of the circle and crosses the...

Do you want to know how to prove that the parallelogram in front of you is actually a rectangle? First, you should know that the formal definition of a rectangle is a parallelogram whose angle is degrees. Additionally, if the diagonals in parallelograms are equal, it is a rectangle.That...

To equalize decimal measures, we will proceed according to the following steps:First step: Identify the larger measure.Second step: Convert the number with the smaller unit of measure to the larger unit of measure.Third step: Compare the numbers that now have the same unit of measure...

What is a repeating decimal? A repeating decimal is a number with a fractional part that, after the decimal point, the digits repeat infinitely, in a periodic manner....

Every three-dimensional body has volume. For example, a ball or pyramid are bodies with volume. The volume of a body is our way of measuring the space that said body occupies in space.For example, let's observe a cube whose length of each of its sides is , like this one:To...

We will say that a function is constant when, as the value of the independent variable increases, the dependent variable remains the same.Let's assume we have two elements , which we will call and , where the following is true: , that is, is located to...

We will solve fraction divisions in the following way:First stepLet's look at the exercise.If there is any mixed number - we will convert it into a fractionIf there is any whole number - we will convert it into a fractionSecond stepWe will convert the division into multiplicationAlso, we will swap...

We will divide the total amount by the denominator of the part, multiply the result obtained by the numerator of the part and obtain the partial amount....

The variation of a function means the rate at which a certain function changes. The rate of variation of a function is also called the slope.According to the mathematical definition, the slope represents the change of the function by increasing the value of by .If the function's graph...

To add fractions, we must find the common denominator simplifying, expanding, or multiplying the denominators. Then, you only need to add the numerators to get the result....

A fraction is actually a division exercise! A result obtained from a division exercise is called a quotient and if it is incomplete, it will appear in the form of a fraction.Remember the rules:The fraction line - symbolizes the division operation.The numerator - symbolizes the number that is being divided...

To subtract fractions, we must find the common denominator by simplifying, expanding, or multiplying the denominators. Then, we only need to subtract the numerators to get the result....

We will solve the multiplication of decimal numbers using the vertical multiplication method.We will proceed in the following order:We will neatly write the multiplication exercise in vertical form – one decimal point under the other decimal point, tenths under tenths, hundredths under hundredths, etc.We will solve the exercise and, for...

There are different types of currencies (different monetary units) in the world. For example, in some countries in Latin America, the peso is used, in others the dollar, and in several European countries, the euro is used.Each of these monetary units is composed of smaller units. For example:In one peso...

The multiplication of fractions is carried out by multiplying numerator by numerator and denominator by denominator, this is the method.In case there is any mixed number - We will convert it into a fraction and then solve according to the learned method.In case there is any whole number - We...

Equivalent EquationsEquivalent equations are different equations whose basically equal to each other, as they share the same variable, in addition to sharing the same solution. That means that if two equations simplify to the same solution when solved, they are equivalent.We can easily move from one equivalent equation to another....

The integer is multiplied by the denominator. The obtained product is then added to the numerator. The final result is placed as the new numerator.Nothing is changed in the denominator.A fraction greater than is a fraction whose numerator is larger than the denominator....

A common denominator is a denominator that will be common and equal for all the fractions in the exercise. We will reach such a denominator by reducing or enlarging the fraction - an operation of multiplication or division. We can arrive at several correct common denominators....

Decreasing functionWhat is a Decreasing Function?A decreasing function is a type of relationship where, as you move to the right on the graph (increasing the xxx-value), the -value gets smaller. It’s like going downhill—the farther you go (the more you increase ), the lower your height (the -value) becomes.We will...

We will perform the same multiplication operation on the numerator and the denominator: the value of the fraction will be preserved.You can expand as many times as you want and by any number....

We will solve addition and subtraction operations of decimal numbers in vertical form, always keeping in mind the following rules: • All the rules that are applicable to the addition and subtraction of whole numbers also apply to decimal numbers. • The decimal points must always be aligned one under...

Regardless of whether the coefficients of the terms are positive or negative, as long as they appear in the style of a trinomial, the exercise will be called "trinomial"....

To add and subtract mixed numbers, we will proceed in 3 steps.The first step:We will convert the mixed numbers into equivalent fractions - fractions with numerator and denominator without whole numbers.The second step:Find a common denominator (usually by multiplying the denominators).The third step:We will only add or subtract the numerators....

We are here to define what a central angle in a circle is and give you tips to remember its definition and properties in the best and most logical way. Before talking about the central angle in a circle, let's take a moment to look at its name - a...

When you come across signs like or , , and even or ,you will know it is an inequality.Inequalities define ranges of possible values rather than single solutions, whether one value is less than, greater than, or equal to another, helping to describe situations where quantities can...

Increasing functionsWhat is an Increasing Function?An increasing function is a type of relationship where, as you move to the right on the graph (increasing the -value), the -value also gets bigger. It’s like climbing a hill—the higher you go (the more you increase ), the more your height (the -value)...

Multiplying a whole number by a fraction and a mixed number is solved in the following steps:The first step:Convert each whole number and mixed number into a similar fraction and rewrite the problem.The second stage:Multiply the numerators and the denominators separately.The multiplication of numerators will be written in the new...

In multiplications: the decimal point moves to the right as many steps as the number has zeros.In divisions: the decimal point moves to the left as many steps as the number has zeros....

Inscribed angle in a circleAn inscribed angle in a circle is an angle whose vertex is on the top of the circle (on the circumference of the circle) and whose ends are chords in a circle.Therefore, if you draw any two chords in a circle, they will meet at the...

When finding an expression with multiplication or an exercise that has only multiplication operations inside a parenthesis and the wholes expression is raised to a certain exponent, we can take the exponent and apply it to each of the terms of the expression or exercise. We must not forget to...

The area of a right triangle is an important subtopic that is repeated over and over again in exercises that include any right triangle.It is calculated by multiplying the two sides that form the right angle (called legs) and dividing the result by 2....

The area of the circle is, in fact, the surface that is "enclosed" within the perimeter of the circumference. It is calculated by raising the radius of the circumference to the second power and multiplying the result by -> . The area of the circle is usually denoted by...

The center of the circumference belongs to subtopics that make up the topic of the circumference and the circle. We use the concept of the center of the circumference to define the circumference itself, as well as to calculate the radius and diameter of each given circumference.The center of the...

When we are presented with exercises or expressions where multiplication of powers with the same base appears, we can add the exponents.The result obtained from adding the exponents will be the new exponent and the original base is maintained.The formula of the rule:It doesn't matter how many terms there are....

A tangent to a circle is a line that touches the circle at one point.Tangent Theorem:1) The tangent to the circle is perpendicular to the radius at the starting point2) Every line perpendicular to the radius at its end is tangent to the circle3) The angle between the tangent and...

When we have an expression raised to a power that, in turn, is raised (within parentheses) to another power, we can multiply the exponents and raise the base number to the result of this multiplication....

When we encounter an expression with a quotient (or division) inside parentheses and the entire expression is raised to a certain exponent, we can take the exponent and apply it to each of the terms in the expression. Let's not forget to maintain the fraction bar between the terms. Formula...

An absolute value is the distance from the zero point, that is, it does not refer to the sum of the number (whether negative or positive), but it focuses on how far it is from the point .The absolute value is symbolized as follows : Generally we can write:...

In any polygon, you can calculate the sum of its internal angles using the following formula:Sum of the internal angles of a polygon: while The number of edges or sides of the polygonSteps to find the sum of the internal angles of a polygon:Count how many sides it...

The formula to calculate the perimeter of a parallelogramYou have probably already realized that it is not necessary to calculate all the edge lengths to find the perimeter.Let's look at the parallelogram :The equal edges are marked with the letters and . Let's note the perimeter of the parallelogram:Now...

Different Types of Rates of Change of a FunctionThe rate of change of a function describes the pace of modification that the variables experience with respect to the change in the variables . The rate of change can be: Constant Rate of Change - fixedDescribes a situation in which...

When we encounter exercises or expressions with terms that have the same base and between them the sign of division or fraction line, we can subtract the exponents. We will subtract the exponent in the denominator from the exponent in the numerator. That is: "exponent of the denominator - exponent...

The intervals where the function is increasing show a certain situation in which the values of and increase together. The intervals where the function is decreasing expose a certain situation in which the value of in a function increases while that of decreases. ...

A bisector is a line segment that passes through the vertex of an angle and divides it into two equal angles.The bisector can appear in a triangle, parallelogram, rhombus and in other geometric figures.For example, a bisector that passes through an angle of degrees will create two angles of...

A right angle is one of the types of angles that we will encounter during engineering studies.A right angle is one that measures . We generally mark it with a small square at the area where it is formed.Right angles can appear in triangles, squares, rectangles, and other geometric shapes...

From time to time, we will come across exercises in which we must use all the properties of powers together. As soon as you have the exercise, try to first get rid of the parentheses according to the properties of powers and then, apply these properties to the corresponding terms,...

Angle notation is the way of naming them to know which angle we are referring to. When reading the data, it is customary to mark the given angles and the angle that needs to be found. In this way, we can obtain a better visual image that allows us to...

To find out when the parabola is positive and when it is negative, we must plot its graph.Then we will look atWhen the graph of the parabola is above the axis, with a positive value, the set is positiveWhen the graph of the parabola is below the ...

If we add a third line that intersects the two parallel lines (those lines that could never cross), we will obtain various types of angles.To classify these angles we must observe if they are:above the line - the pink partbelow the line - the light blue partto the right of...

In this article, we will briefly learn everything necessary about triangles and also practice with some exercises!Let's get started!...

First step: Let's reduce the fractions if possible.Second step:Let's convert the mixed numbers into fractions....

The denominator is the bottom number of a fraction and represents the whole in its entirety.For example:...

What is the numerator? The numerator is the top number of a fraction and represents the portion within the whole part....

Raising any negative number to an even power will result in a positive outcome.When is even:...

An equation is a type of exercise that carries a sign which, on each side of the sign, that is, in each member of the equation there is an algebraic expression.An algebraic expression can be anything -> just a number, just an unknown or well, an exercise with number...

There is no root of a negative number since any positive number raised to the second power will result in a positive number....

The perimeter indicates the distance we will walk if we start from a certain point, complete a full lap, and return exactly to the starting point.For example, if we are asked what the perimeter of the waist is, we will take a tape measure and measure the perimeter from a...

In this article, we will teach you the basics of everything you need to know about mixed numbers.If you wish to delve deeper into a specific topic, you can access the corresponding extensive article....

The decimal number represents, through the decimal point (or comma in certain countries), a simple fraction or a number that is not whole.The decimal point divides the number in the following way:You can read more in the assigned extended article...

Learn the multiplication tables thoroughly and follow these rules:...

In this article, we will learn how to perform mathematical calculations with fractions.More reading material:Addition of fractionsSubtraction of fractionsMultiplication of fractionsDivision of fractionsComparison of fractions...

The vertex of the parabola indicates the highest or maximum point of a sad-faced parabola, and the lowest or minimum point of a happy-faced parabola....

The rectangular prism is a three-dimensional figure composed of rectangles....

Prime factorization (or prime decomposition) consists of breaking down a certain number into prime numbers, called factors, whose product (multiplication) results in the original number....

What's its name? Rhombus, kite, or diamond? ;)...

The isosceles trapezoid is, in fact, a trapezoid (that is, a four-sided polygon with two of them - the bases - being parallel), with two of its sides being equivalent and with its base angles of equal magnitude....

More than once we have heard the teacher ask in class: "Who knows how to solve a quadratic equation without the formula?" We looked around to see who knew the answer, "Do you know what a trinomial is?" The teacher continued asking. We doubted and thought about what word this...

The average is, in fact, a number that represents a group of numbers. It is the average, its center, therefore, it represents them.When we ask, for example, what is the average height of the third grade B students, in reality, we are asking what is the height that would represent...

A number is divisible by if the units digit is even - that is, it divides by without a remainder....

Divisibility criteria for , and ...

A prime number is a natural number that is divisible only by itself and by ....

The process of completing the square is a way to solve a quadratic equation. This procedure converts an equation written in the standard form of the quadratic function into an expression with a variable squared, as in the following example: where and are parameters....

This form is called factored because it uses the factors of a multiplication.With this form, we can easily identify the points of intersection of the function with the axis.The factored form of the quadratic function looks like this:...

The vertex form allows us to identify, very easily, the vertex of the parabola and hence its name.The vertex form of the quadratic function is:...

The standard form of the quadratic function is:For example:...

Family of Parabolas Combination of Horizontal and Vertical ShiftIn this quadratic function  determines the amount of steps and the vertical direction in which the function will shift - upwards or downwards. determines the amount of steps and the horizontal direction in which the function will shift - to the right or to...

Family of Parabolas In this family, we have a slightly different quadratic function that shows us, very clearly, how the parabola shifts horizontally. indicates the number of steps the parabola will move horizontally, to the right or to the left. If is positive: (there is a minus sign in...

Family of Parabolas : Vertical ShiftThe basic quadratic function  with the addition of yields the function  The meaning of is the vertical shift of the function upwards or downwards. If is positive: the function will rise by the number of steps shown in . If is...

The functions Properties of the function:The most basic quadratic function Minimum, happy face function, its vertex is The axis of symmetry of this function is .The function's interval of increase: The function's interval of decrease: Set of positivity: Every except . Set of negativity: None....

A quadrilateral whose sides (or edges) are all equal and all its angles are also equal, is a square.Furthermore, a square is a combination of a parallelogram, a rhombus, and a rectangle.Therefore, the square has all the properties of the parallelogram, the rhombus, and the rectangle. Square ...

You will be able to determine that the parallelogram is a rhombus if at least one of the following conditions is met:If in the parallelogram there is a pair of adjacent equal sides - it is a rhombus.If in the parallelogram the diagonals bisect each other, forming angles of ...

The exterior angle of a triangle is the one that is found between the original side and the extension of the side. The exterior angle is equal to the sum of the two interior angles of the triangle that are not adjacent to it.It is defined as follows:...

The key to adding or subtracting algebraic fractions is to make all denominators equal, that is, to find the common denominator. To do this, we will need to factorize according to the different methods we have learned.Action steps:We will factorize all the denominators we have.We will note the common denominator...

Algebraic fractions are fractions with variables....

The notation of a function actually refers to determining the "name" of the function.It is customary to symbolize a function using letters from the Latin alphabet when the two most common notations are:(Of course, similar notations can also be used).The - inside parentheses expresses that it is an independent variable...

A weighted average is an average among numbers with different weights. Each number has its own weight and, therefore, will affect the weighted average. Try replacing the word weight with the word importance and in this way its meaning will be better understood. The numbers are of different importance. One...

The regular hexagon belongs to the family of regular polygons. It is a polygon in which all sides, and all angles, are equal to each other. By its name, we can understand that it is a geometric figure with different sides. The sum of its internal angles equals ...

Perpendicular lines are vertical lines that form a right angle between them, that is, an angle of degrees. Perpendicular lines appear in many geometric shapes, such as a rectangle, a square, a right triangle, and others. ...

The "absolute value" may seem complicated to us, but it is simply the distance between a given number and the figure . ...

Fractions do not influence the order of operations, therefore, you should treat them like any other number in the exercise.The correct order of mathematical operations is as follows:ParenthesesMultiplications and divisions in the order they appear in the exerciseAdditions and subtractions in the order they appear in the exercise...

Comparing decimal numbers is done using the system: Digit-by-digit analysis...

The addition and subtraction of real numbers are based on certain key principles. All principles will be explained using two real numbers, but certainly, the numbers in the exercise do not influence the method of resolution, therefore, these principles can be applied to any number in the exercise.When we have...

Congruent rectangles are those that have the same area and the same perimeter. Let's look at an exercise as an example: Given the rectangles and , as described in the following scheme:Observe the data that appears in the scheme and determine if they are congruent rectangles.In the first rectangle we see...

The exponent implies the number of times the base of the power must multiply itself. In order for the base of the power to know how many times it should multiply itself, we must look at the exponent. The exponent denotes the power to which the base must be raised,...

The division of whole numbers within parentheses where there is a division refers to the situation in which we must carry out the mathematical operation of dividing a whole number by the result of dividing two elements, that is, by their quotient.For example:There are two ways to solve this type...

The distributive property of multiplication allows us to break down the highest term of the exercise into a smaller number. This simplifies the multiplication operation and we can solve the exercise without the need to use a calculator....

An scalene triangle is a triangle that has all its sides of different lengths....

An obtuse triangle is a triangle that has one obtuse angle (greater than degrees and less than degrees) and two acute angles (each of which is less than degrees). The sum of all three angles together is degrees....

An acute triangle has all acute angles, meaning each of its three angles measures less than degrees and the sum of all three together equals degrees. ...

Simplifying Like Terms in an EquationWhen solving equations, simplifying like terms—terms with the same variable and exponent—makes the equation easier to solve by consolidating similar elements. Simplify the like terms in an equation involves combining the elements that belong to the same group. In other words: in all first-degree equations...

Multiplying or Dividing Both Sides of the EquationSometimes when solving equations, we may encounter variables with coefficients, which we need to remove to isolate the variable and find its value. Exactly for those cases, and many more, we have the ability to multiply or divide both sides of the equation...

This method allows us to add or subtract the same element from both sides of the equation without changing the final result, that is, the outcome of the equation will not be affected by the fact that we have added or subtracted the same element from both sides.Let's see what...

In previous articles, we have studied real numbers and the grouping of terms, as well as the order of mathematical operations with parentheses. In this article, we move forward and combine these topics in order to understand when and how we can eliminate parentheses in real numbers....

In previous articles, we have seen what is the order of operations for addition, subtraction, multiplication, and division and also the order we must follow when there are exponents.When the exercise we need to solve includes parentheses, we always (always!) start with the operation contained within them.ParenthesesExponents and rootsMultiplications and...

As part of combined operations, we learned that parentheses always come first.Once solved, we can begin to simplify powers (or roots).After simplifying them, we can continue solving the exercise according to the order of basic operations:Firstly, the multiplications and divisions, and lastly, the additions and subtractions.Let's refresh the order of...

Two numbers are multiplicative inverses when their product results in .For example: and are multiplicative inverses because ...

Abbreviated multiplication formulas will be used throughout our math studies, from elementary school to high school. In many cases, we will need to know how to expand or add these equations to arrive at the solution to various math exercises.Just like other math topics, even in the case of abbreviated...

Some of you may know the radius as a "dial". Either way, the meaning is identical with the same characteristics. So, what is the radius? It is a specific segment that connects the center of a circle with a particular point on the circumference ....

First, we must differentiate the following two concepts to avoid confusion:Average speedAverage velocityAt first glance, this looks like the same term, but in practice, it is not. Average speed asks you to know what is the general-classical average of the speed at which several drivers were traveling:Example:Ivan traveled at ...

We can add angles and get the result of their sum, and we can also subtract them to find the difference between them. Even if the angles don't have any numbers, we'll learn how to represent their sum or difference and arrive at the correct result....

Definition: Angles are created at the intersection between two lines. As seen in the following illustrationThe angle in the illustration is called . We could also call it angle . The important thing is that the middle letter is the one at the intersection of the lines.For example, in this...

On one hand, functions are a fairly abstract concept, but on the other hand, they are very useful in many areas of mathematics. The topic of functions dominates many fields, including algebra, trigonometry, differential and integral calculus, and more. Therefore, it's important to understand the concept of functions, so that...

An obtuse angle is an angle that measures more than degrees, but less than degrees.Acute angles can appear in triangles, parallelograms, and other geometric shapes.Obtuse Angle ...

Domain of a FunctionWhat is the Domain of a Function?The domain of a function includes all those values of (independent variable) that, when substituted into the function, keep the function valid and defined. In simple terms, the domain tells us what numbers we are allowed to plug into the...

A value table is the "preparatory work" that we are often asked to do before producing a graphical representation. Therefore, it is an inseparable part of the subject of graphs in general and the topic of functions in particular....

Long division is a method used to divide large numbers by breaking down the process into a series of easier steps, dealing with one digit at a time. This technique is especially useful for dividing numbers that don’t divide evenly.How to use Long Division?We'll go step by step, dividing one...

To locate fractions on the number line, we will carry out several steps....

MidsegmentThe midsegment is a segment that connects the midpoints of 2 sides. It's very simple to remember the meaning of this term since the word "middle" already tells us that it is about the midpoint, so when we come across the concept of "midsegment" we'll remember that it connects the...

Algebraic MethodAlgebraic Method is a general term for various tools and techniques that will help us solve more complex exercises in the future. It is mostly concern about using algebraic operations to isolate variables and solve equations. This approach is fundamental for solving equations in various mathematical contexts.Distributive PropertyThis property...

How do you simplify fractions? Or, how do you reduce fractions?In most cases, when fractions are introduced to students as a new topic in the classroom, the initial reaction is: "Here's another complex subject we have to deal with." But then, reactions change and fractions are seen as a kind...

The height of a triangle is the segment that connects a vertex to the opposite side such that it creates a 90-degree angle.In every triangle, there are three heights, as there are three vertices from which the height can be calculated relative to the side that is opposite to each...

(square centimeter), (square meter), (square kilometer).These units are different, but they are related:Understanding the relationship between these units is key, but there's no need to memorize it—we can quickly calculate it when needed.Let's say we want to calculate how many are in . We’ll draw a...

Compared to other geometric figures, the rectangle is one of the simplest to work with.One of the most frequent questions that comes up in exams is related to how to calculate the area of the rectangle.Before we focus on it, let's do a brief review.The formula for calculating the area...

A rectangle is a quadrilateral with two pairs of parallel opposite edges (sides), the angles of which all equal 90 degrees.The pairs of sides in a rectangle are opposite, equal, and parallel.Each of the angles in a rectangle are equal to 90 degrees.The diagonals of a rectangle are equal.The diagonals...

When we study the order of mathematical operations we come across the terms division bar and fraction bar, but what do they mean and why are they so special?First of all, we must remember that the fraction bar—or vinculum—is exactly the same as a division. is the same as...

In mathematics, a neutral element is an element that does not alter the rest of the numbers when we perform an operation with it.Neutral Element - AdditionWith addition, is considered a neutral element because it does not modify the number to which it is added.Neutral Element - MultiplicationIn multiplication,...

The numbers and have some special characteristics when performing basic operations like addition, subtraction, multiplication, and division—including combined calculations.In this article we will learn what they are and why they are important....

As we have learned in previous lessons, when working with combined operations the order of the basic operations must be followed in order to get the correct result. However, before performing these the parentheses and then the roots and powers must first be solved.Roots are very important in mathematical calculations....

The rules for the order of operations in an addition and subtraction exercise are quite simple.In exercises with combined operations in which there are also multiplications and divisions, the order of operations is as follows:ParenthesesPowers and rootsMultiplications and divisionsAddition and subtraction...

The simplification of expressions consists of creating an equivalent expression written in a shorter and simpler way in which we combine all of the similar terms (collecting like terms).For example, the expression:After having simplified it, it would be:What we have done is created two groups of numbers and variables: and...

In previous articles, we have talked about what an algebraic expression is and how to get the numerical value of algebraic expressions. Today, we will cover equivalent expressions.Equivalent expressions are two or more algebraic expressions that represent the same value. They may have a different structure, but their numerical value...

An algebraic expression is a combination of constant numbers (or 'integers'), unknown variables represented by letters, and basic operations. When we assign numerical values to each of the unknown variables, we can reduce the expression to a numerical value.If you are unsure about these terms, you can click on the...

The Pythagorean Theorem can be formulated as follows: in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs.In the right triangle shown in the image below, we use the first letters of the alphabet to indicate its sides: and ...

The order of operations is a convention used to determine which operations are performed first. In every math exercise that combines more than one operation (addition, subtraction, multiplication, division, etc.), each operation must be performed in a specific order:ParenthesesPowers and RootsMultiplication and division (from left to right)Addition and subtraction (from...

For example:One way to solve this exercise will be to remove the parentheses. To do this, we must remember the rule that states that, in order to remove the parentheses, we must divide the whole number by each of the terms of the multiplication operation in parenthese.That is, in our...

Subtraction of whole numbers with subtractions in parentheses refers to a situation where we perform the mathematical operation of subtraction on the difference of some terms that are in parentheses.For example:One way to solve this exercise will be to distribute the parentheses. To do this, we must remember that according...

Subtraction of whole numbers with addition in parentheses refers to a situation where we must perform the mathematical operation of subtraction on the sum of some terms that are in parentheses.In this case, we must remember that the subtraction will be performed on each and every term separately.The rule is...

The distributive property of division allows us to break down the first term of a division expression into a smaller number. This simplifies the division operation and allows us to solve the exercise without a calculator.When using the distributive property of division, we begin by breaking down the number being...

The associative property of multiplication allows us to multiply two factors and then multiply the product by the third factor, even if they're not in order from left to right.We can use this property in three ways:1. We start by multiplying the first and the second factors, solving, and then...

The associative property of addition allows us to group two addends and then add the third addend to the result.We can use this property in three ways:1. We start by adding the first and the second addend, solve the sum and add the third addend to the result.2. We start...

The associative property tells us that that we can change the grouping of factors (in multiplication) or addends (in addition) in an expression without changing the end result.Typically, we use parentheses to associate, since they come first in the order of operations (PEMDAS).For example:The expressionCan be associated as...

The commutative property of multiplication tells us that changing the order of factors in an expression doesn't change the answer - even if there are more than two!Like the commutative property of addition, the commutative property of multiplication helps us simplify basic expressions, algebraic expressions and more.We can define the...

Exponents are a shorthand way of telling us that a number is multiplied by itself.The number that is multiplied by itself is called the base. The base is the larger number on the left.The smaller number on the right tells us how many times the number is multiplied by itself....

A cube is a type of cuboid in which all three dimensions (length, width and height) are identical. All cubes are made up of of six identical squares.To find the volume of a cube we must go through the same steps as to find the volume of an cuboid, that...

The rectangular cuboid, or just cuboid, is a three-dimensional shape that consists of six rectangles. Each rectangle is called a face. Every rectangular cuboid has six faces (The top and bottom faces are often called the top and bottom bases of the rectangular cuboid). It is important to understand that...

Quadratic equations (also called second degree equations) contain three numbers called parameters:Parameter represents the position of the vertex of the parabola on the axis. A parabola can have a maximum vertex, or a minimum vertex (depending on if the parabola opens upwards or downwards).Parameter represents the position...

To find the area of a trapezoid, you need the following three pieces of information:The length of base oneThe length of base twoThe height between the two basesThe formula to find the area of a trapezoid is as follows:The sum of the bases multiplied by the height and then divided...

Parallel lines are lines that belong to the same plane (are coplanar) and never meet (do not intersect).Let there be two parallel lines and as shown below.If we state the following:The straight line is parallel to the straight line we can say the same thing using mathematical...

When a problem is presented to us in writing, we can convert it into mathematical language (also called algebraic language) by transforming it into an algebraic expression. But what are algebraic expressions?Variable: This is a letter that represents a numerical value, for example or . This letter refers to...

Mathematical sequences are a group of terms with a certain rule that dictates a certain operation must be performed and repeated in order to get from one term to the next.The operation can be addition, subtraction, multiplication, division, or any other mathematical operation.For example, the following is a basic numerical...

If there is a relationship between the elements of a sequence, the recurrence relation would be the rule that connects them. It is possible to formulate the recurrence relation and use it to find the value of each of the elements of the set according to the position it occupies.For...

Solving algebraic equations is made easier by understanding some basic rules and properties. A few examples of properties that we will learn to use in the seventh grade are: the distributive, associative and commutative properties. These properties get learned and relearned throughout our time in school, each time adding new...

The extended distributive property allows us to solve exercises with two sets of parentheses that are multiplied by eachother.For example: To find the solution, we will go through the following steps:Step 1: Multiply the first term in the first parentheses by each of the terms in the second parentheses.Step 2:...

What is the distributive property?The distributive property is a rule in mathematics that says that multiplying a number by the sum of two or more numbers will give us the same result as multiplying that number by the two numbers separately and then adding them together.For example, 4x4 will give...

The Parabola This function is a quadratic function and is called a parabola.We will focus on two main types of parabolas: maximum and minimum parabolas....

To convert a decimal number to a simple fraction we will ask ourselves how the decimal number is read. If we use the word tenths, we will place in the denominator If we use the word hundredths, we will place in the denominator If we use the word...

The decimal number is a way to represent a simple fraction or a number that is not whole.The decimal point (or decimal comma in some areas) divides the number in the following way:For example, when checking a fever, on the thermometer there is a number like or .The point...

In this article, we will learn what area is, and understand how it is calculated for each shape, in the most practical and simple way there is. Shall we start?...

Formula to calculate the area of an equilateral triangle:...

Formula to calculate the area of a scalene triangle:...

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What is ratio?The ratio describes the "relationship" between two or more things.The ratio connects the given terms and describes how many times greater or smaller a certain magnitude is than another.Let's see an example from everyday life:When asked in a class, what is the ratio between boys and girls, it...

Inverse proportionality describes a situation in which, when one term is multiplied by a certain number of times, the second term is decreased by the same number of times. This also occurs in reverse, if one term decreases by a certain number of times, the second term increases by the...

Direct proportionality indicates a situation in which, when one term is multiplied by a certain amount, the same exact thing happens to the second term.In the same way, when one term is divided by a certain amount, the same exact thing happens to the second term.The ratio between both magnitudes...

Division in a given ratio means splitting a total quantity into parts that maintain a specific proportional relationship, based on the ratio provided. In a division according to a given ratio, we will have a defined quantity that we must divide according to said ratio. The process ensures that the...

To easily solve ratio problems and to gain a better understanding of the topic in general, it is convenient to know about equivalent ratios.Equivalent ratios are, in fact, ratios that seem different, are not expressed in the same way but, by simplifying or expanding them, you arrive at exactly the...

The axis of symmetry in a parabola is the axis that passes through its vertex in such a way that if we folded the right side over the left side, both sides would appear joined. Let's see it in an illustration:To find the axis of symmetry, we must locate the...

Where , since if the coefficient does not appear then it would not be a quadratic function.The graph of a quadratic function will always be a parabola.Example 1:It is a quadratic or second-degree function because its largest exponent is .Example 2:This is not a second-degree function because, although it...

The diagonals of a rhombus have 3 properties that we can use without having to prove them:The diagonals of a rhombus intersect. (not only do they intersect, but they do so exactly at the midpoint of each one).The diagonals of a rhombus are perpendicular, forming a right angle of ...

The area of the kite can be calculated by multiplying the lengths of the diagonals and dividing this product by ....

Factorization allows us to convert expressions with elements that are added or subtracted into expressions with elements that are multiplied....

Pythagorean Theorem in an OrthohedronThe orthohedron or cuboid is a rectangular prism, a three-dimensional figure, that is, it has length, width, and height (or depth). In addition, the angles between the different planes are right angles, which allows us to make use of the Pythagorean theorem to calculate the length...

A linear function describes the relationship between and .Therefore, we can represent all sorts of different phenomena in life with the help of the linear function.The representation of phenomena with the help of linear functions is expressed in mathematics in word problems, using graphs of the functions.Thus, we can...

Positivity and Negativity of a Linear FunctionThe function is positive when it is above the axis when The function is negative when it is below the axis as When we are asked what the domains of positivity of the function are, we are actually being asked in which...

Finding a linear equation is actually about graphing the linear function using or .We can find the linear equation in ways:Using a point on the line and the slope of the line.Using two points that lie on the line.Using the graph of the function itself.Using parallel lines, that...

The linear function actually represents a graph of a straight line that has a point of intersection with the vertical axis. represents the slope.When is positive, the slope is positive: the line goes upwards.When is negative, the slope is negative: the line goes downwards.When , the...

The graphical representation of a function that represents direct proportionality is actually the ability to express an algebraic expression through a graph.As it is a direct proportionality, the graph will be of a straight line.A function that represents direct proportionality is a linear function of the family .The graphical representation...

A linear function, as it is called, is an algebraic expression that represents the graph of a straight line.When we talk about functions, it's important to highlight that the graphs of functions are represented in an axis system where there is a horizontal axis and a vertical axis .Linear...

SummaryRatio, Proportion, and ScaleThe ratio between terms describes how many times greater or smaller a certain magnitude is than the other.Proportion is a constant relationship or ratio between different magnitudes.Scale is the proportionality between the real dimensions of something and those of the scheme that represents it....

We can identify that the square in front of us is a parallelogram if at least one of the following conditions is met:If in a square each pair of opposite sides are parallel to each other, the square is a parallelogram.If in a square each pair of opposite sides are...

The diagonals of an isosceles trapezoid have the same length. This theorem also holds true in reverse, meaning, we can determine that a certain trapezoid is isosceles if we know that its diagonals are equal.You can use this theorem in the exam as you see it and you will not...

To solve equations through factorization, we must transpose all the elements to one side of the equation and leave on the other side. Why? Because after factoring, we will have as the product....

When we want to multiply or divide algebraic fractions, we will use the same tools that we use for the multiplication or division of common fractions with some small differences.Steps to carry out for the multiplication of algebraic fractions :Let's try to extract the common factor. This can be the...

When we have equal numbers or with a common denominator in the numerator and in the denominator, in certain cases, we can simplify fractions.Often we will encounter an algebraic fraction in which the numerator and the denominator can be simplified. For example, this equation:is a fraction that we can simplify....

The formulas for shortened multiplication will help us convert expressions with terms that have among them signs of addition or subtraction into expressions whose terms are multiplied. The formulas for contracted multiplication are:...

Factorization is the main key to solving more complex exercises than those you have studied up to today. Factorization helps to solve different exercises, among them, those that have algebraic fractions. In exercises where the sum or difference of their terms equals zero, factorization allows us to see them as...

Powers are the number that is multiplied by itself several times. Each power consists of two main parts: Base of the power: The number that fulfills the requirement of duplication. The principal number is written in large size.Exponent: the number that determines how many times the power base needs to be...

To solve systems of two linear equations with two unknowns with the equating method, we must arrive at a situation in which one of the coefficients is equal or opposite in the same unknown in the two equations.How do we do this?We will arrange the equation in such a way...

To solve with the substitution method a system of two linear equations with two unknowns we will have to substitute one of the unknowns in some equation and thus obtain an equation with only one unknown.How do we do it?Choose the equation in which you can easily isolate one of...

A system of linear equations is, in fact, a set of conditions that must be satisfied by specific unknowns, the solution for the system of equations is then based on finding the and the that agree with both the first equation and the second equation.These questions can be...

An equation that has two variables: and . To solve a linear equation that has two variables, we must find a pair of values for and for that preserve the equation. How will we do it?Try to isolate one variable, whichever you prefer, then leave it alone...

A linear equation is an equation of the type:A system of two linear equations with two unknowns is a pair of adjacent linear equations or written one below the other, either within braces or without graphic signs.To solve a system of equations, several steps must be taken:Isolate the variables in...

To demonstrate the similarity between triangles it is not necessary to show again and again the relationship between the three pairs of sides and the equivalence between all the corresponding angles. This would require too much unnecessary work.There are three criteria by which we can see the similarity between triangles:Angle...

Congruence in geometry refers to two figures that have the exact same shape and size, meaning they can perfectly overlap when placed on top of one another.There are 4 criteria to determine that two triangles are congruent. In this article, we will learn to use the fourth criterion of congruence:Fourth...

Understanding the combination of powers and roots is important and necessary.First property: Second property:Third property: Fourth property: Fifth property:  ...

When we encounter an exercise in which there is a root applied to another root, we will multiply the order of the first root by the order of the second and the order obtained (the product of both) will be raised as a root in our number (as generally power...

When we find a root that is in the complete quotient (in the complete fraction), we can break down the factors of the quotient: the numerator and the denominator and leave the root separated for each of them. We will not forget to leave the division symbol: the dividing...

When we encounter a root that encompasses the entirety of the product, we can decompose the factors of the products and leave a separate root for each of them. Let's not forget to leave the multiplication sign between the factors we have extracted.Let's put it this way:...

A root is the inverse operation of a power.It is denoted with the symbol and is equal to a power of .If a small number appears on the left, it will be the order of the root....

The factorization we do by extracting the common factor is our way of modifying the way the exercise is written, that is, from an expression with addition to an expression with multiplication.For example, the expression is composed of two terms and a plus sign. We can factor it by excluding...

The trapezoid is considered one of the most intimidating shapes for students, therefore we have decided to provide a summary of the general idea behind the trapezoid and explain its properties to them and introduce some types of trapezoids....

In fact, estimation allows us to guess (hence the redundancy) the supposed result, without performing the exact calculation. That is, in certain cases, we don't need to know the solution precisely, a rough idea is sufficient to solve a particular mathematical problem.Sometimes we are asked to compare mathematical expressions, draw...

The formula for calculating the area of a triangle of any type:height times base divided by .How to find the area of a triangle:...

There are a few logarithmic laws worth knowing to make solving problems easier. The following laws are the main rules you will use. It should be noted that the letters a, m, n must be positive real numbers for these laws to be valid.Logarithmic LawsConstant Values:It can be automatically determined...

Parallelogram is a four-sided polygon (quadrilateral) where opposite sides are parallel and equal in length. A key feature of parallelograms is that they have two sets of parallel lines, which gives them their name. Examples of parallelograms include squares, rectangles, and rhombuses, which are all specific types of parallelograms with...

Every geometric problem is based on data, and the solution is divided into several different questions. One of the most popular questions, and the one most likely to appear on your test, is the question about the area of a rhombus. As is well known, a rhombus consists of ...

In geometry, a deltoid is defined as a quadrilateral consisting of isosceles triangles that share a common base.So, what is the identification of a deltoid in the family of quadrilaterals?A quadrilateral that has 2 pairs of equal adjacent sidesExample:If given : Then: is a deltoid by definition.2 isosceles...

What is a percentage?A percentage is a way to define a part, or fraction of a total.When we talk about percentages, we should always ask ourselves the following: "the percentage of what?". Saying 50% without specifying 50% of what, makes no sense. However, if we say "the of "...

Congruent triangles are identical triangles.That means in triangles whose angles and sides are equal, their area and perimeter will also be equal.But keep in mind that this case is different from when triangles are similar, that is, when the angles are equal but the side lengths are different in the...

What are those mysterious square roots that often confuse students and complicate their lives? The truth is that to understand them, we need to grasp the concept of the inverse operation....

Exponents are a way to write the multiplication of a term by itself several times in a shortened form.The number that is multiplied by itself is called the base, while the number of times the base is multiplied is called the exponent.... (n times)For example: is the base, while ...