The Quadratic Function - Examples, Exercises and Solutions

In fact, a quadratic equation is composed as follows:

y = ax²-bx-c

That is,

a is the coefficient of x², in this case 2.
b is the coefficient of x, in this case 5.
And c is the number without a variable at the end, in this case 6.

Here we have a quadratic equation.

A quadratic equation is always constructed like this:

$y = ax²+bx+c$

Where a, b, and c are generally already known to us, and the X and Y points need to be discovered.

Firstly, it seems that in this formula we do not have the C,

Therefore, we understand it is equal to 0.

$c = 0$

a is the coefficient of X², here it does not have a coefficient, therefore

$a = 1$

$b= 10$

is the number that comes before the X that is not squared.

The quadratic equation in the problem is already arranged (meaning all terms are on one side and 0 on the other side), so let's proceed to answer the question asked:

The question asked in the problem - What is the value of the coefficient$a$ in the equation?

Let's recall the definitions of coefficients in solving quadratic equations and the roots formula:

The rule states that the roots of an equation of the form:

$ax^2+bx+c=0$are:

$x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

That is the coefficient $a$is the coefficient of the quadratic term (meaning the term with the second power)- $x^2$Let's examine the equation in the problem:

$$$-x^2+7x-9 =0$

Let's remember that the minus sign before the quadratic term means multiplication by: $-1$ , therefore- we can write the equation as:

$-1\cdot x^2+7x-9 =0$

meaning- the number that multiplies the $x^2$, is $-1$ therefore we identify that the coefficient of the quadratic term is the number $-1$,

Therefore the correct answer is A.

The quadratic equation of the given problem is already arranged (that is, all the terms are found on one side and the 0 on the other side), thus we approach the given problem as follows;

In the problem, the question was asked: what is the value of the coefficient$b$in the equation?

Let's remember the definitions of coefficients when solving a quadratic equation as well as the formula for the roots:

The rule says that the roots of an equation of the form

$ax^2+bx+c=0$are :

$x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

That is the coefficient$b$is the coefficient of the term in the first power -$x$We then examine the equation of the given problem:

$$$3x^2+8x-5 =0$That is, the number that multiplies

$x$ is

$8$Consequently we are able to identify b, which is the coefficient of the term in the first power, as the number$8$,

Thus the correct answer is option d.

The quadratic equation of the given problem has already been arranged (that is, all the terms are on one side and 0 is on the other side) thus we can approach the question as follows:

In the problem, the question was asked: what is the value of the coefficient$c$in the equation?

Let's remember the definition of a coefficient when solving a quadratic equation as well as the formula for the roots:

The rule says that the roots of an equation of the form

$ax^2+bx+c=0$are:

$x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

That is the coefficient
$c$is the free term - and as such the coefficient of the term is raised to the power of zero -$x^0$(Any number other than zero raised to the power of zero equals 1:

$x^0=1$)

Next we examine the equation of the given problem:

$$$3x^2+5x=0$Note that there is no free term in the equation, that is, the numerical value of the free term is 0, in fact the equation can be written as follows:

$3x^2+5x+0=0$and therefore the value of the coefficient$c$ is 0.

Hence the correct answer is option c.

Let's recall the general form of a quadratic function:

$y=ax^2+bx+c$

Let's examine the given function in the problem:

$y=6x-6x^2+3$$$

Note that in the general form of the quadratic function mentioned above, the terms are arranged from the highest power (which is the quadratic term - power of 2) to the lowest power (which is the free term - power of 0),

Therefore, to make it easier to identify the coefficients, we'll use the commutative property of addition and rearrange the terms of the quadratic function so they are written from highest to lowest power:

$y=6x-6x^2+3 \\ y=-6x^2+6x+3$

We can then identify that the coefficient of the quadratic term, meaning the coefficient of the term with power two: $a$ is $-6$ We'll continue and identify that the coefficient of the term with power one: $b$ is $6$ and finally we'll identify that the coefficient of the term with power 0, meaning the free term: $c$ is $3$

To summarize, the coefficients in the given function are:

$a=-6,\hspace{4pt}b=6,\hspace{4pt}c=3$

Therefore, the correct answer is answer A.

Note:

The coefficient $c$is the free term - and we said before that it's the coefficient of the term with power zero - $x^0$this is because any number different from zero raised to the power of zero equals 1:

$x^0=1$, and therefore we could write the general form of the function above as:

$y=ax^2+bx+c \\ \downarrow\\ y=ax^2+bx+c\cdot1 \\ \downarrow\\ y=ax^2+bx^1+c x^0$

meaning, $c$ is the coefficient of the term with power 0.