Difference of squares - Examples, Exercises and Solutions

We use the abbreviated multiplication formula:

$4-x^2=0$

We isolate the terms and extract the root:

$4=x^2$

$x=\sqrt{4}$

$x=\pm2$

Let's solve the equation. First, we'll simplify the algebraic expressions using the difference of squares formula:

$(a+b)(a-b)=a^2-b^2$We'll apply this formula and expand the parentheses in the expressions in the equation:

$(x+3)(x-3)=x^2+x \\ x^2-3^2=x^2+x \\ x^2-9=x^2+x$We'll continue and combine like terms. After moving terms around, we can see that the squared term cancels out, therefore it's a first-degree equation, which we'll solve by isolating the variable term on one side and dividing both sides of the equation by its coefficient:

$x^2-9=x^2+x \\ -9=x\\ \downarrow\\ \boxed{x=-9}$Therefore, the correct answer is answer B.

First we rearrange the equation and set it to 0

$4x^2-3-6=0$

$4x^2-9=0$

We then apply the shortcut multiplication formula:

$4(x^2-\frac{9}{4})=0$

$x^2-(\frac{3}{2})^2=0$

$(x-\frac{3}{2})(x+\frac{3}{2})=0$

$x=\pm\frac{3}{2}$

$$The equation in the problem is:

$x^2+10x+50=-4x+1$First, we identify that the equation is quadratic (and this is because the quadratic term in it does not cancel out), therefore, we will simplify the equation by moving all terms to one side and combine thelike terms:

$x^2+10x+50=-4x+1 \\ x^2+10x+4x+50-1 =0 \\ x^2+14x+49 =0 \\$

We want to solve this equation using factorization.

First, we'll check if we can find a common factor, but this is not possible, since there is no multiplicative factor common to all three terms on the left side of the equation.

We can factor the expression on the left side using the quadratic factoring formula for a trinomial, however, we prefer to factor it using the trinomial factoring methodl:

Note that the coefficient of the quadratic term (the term with the second power) is 1, and therefore we can try to perform factoring according to the quick trinomial method:

But before we do this in the problem - let's remember the general rule for factoring with the quick trinomial method:

The rule states that for the algebraic quadratic expression:

$x^2+bx+c$We can find a factorization to the form of a product if we can find two numbers $m,\hspace{4pt}n$such that the conditions (conditions of the quick trinomial method) are met:

$\begin{cases} m\cdot n=c\\ m+n=b \end{cases}$If we can find two such numbers $m,\hspace{4pt}n$then we can factor the general expression mentioned above into the form of a product and present it as:

$x^2+bx+c \\ \downarrow\\ (x+m)(x+n)$which is its factored form (product factors) of the expression,

Let's return now to the equation in the problem that we received in the last stage after arranging it:

$x^2+14x+49 =0$Note that the coefficients from the general form we mentioned in the rule above:

$x^2+bx+c$are:$\begin{cases} c=49 \\ b=14 \end{cases}$Don't forget to consider the coefficient together with its sign.

Let's continue - we want to factor the expression on the left side into factors according to the quick trinomial method, above, so we'll look for a pair of numbers $m,\hspace{4pt}n$ that satisfy:

$\begin{cases} m\cdot n=49\\ m+n=14 \end{cases}$We'll try to identify this pair of numbers using our knowledge of the multiplication table, we'll start from the multiplication between the two required numbers $m,\hspace{4pt}n$ that is - from the first row of the pair of requirements we mentioned in the last stage:

$m\cdot n=49$We identify that their product needs to give a positive result, and therefore we can conclude that their signs are identical.

Next, we'll refer to the factors (integers) of the number 49, and from our knowledge of the multiplication table we can know that there are only two possibilities for such factors: 7 and 7, or 49 and 1, as we previously concluded that their signs must be identical, a quick check of the two possibilities for the second condition:

$m+n=14$ will lead to a quick conclusion that the only possibility for fulfilling both of the above conditions together is:

$7,\hspace{4pt}7$That is:

$m=7,\hspace{4pt}n=7$(It doesn't matter which one we call m and which one we call n)

It is satisfied that:

$\begin{cases} \underline{7}\cdot \underline{7}=49\\ \underline{7}+\underline{7}=14 \end{cases}$ From here - we understood what the numbers we are looking for are and therefore we can factor the expression on the left side of the equation in question and present it as a product:

$x^2+14x+49 \\ \downarrow\\ (x+7)(x+7)$

In other words, we performed:

$x^2+bx+c \\ \downarrow\\ (x+m)(x+n)$

If so we factored the quadratic expression on the left side of the equation into factors using factoring according to the quick trinomial method, and the equation is:

$x^2+14x+49=0 \\ \downarrow\\ (x+7)(x+7)=0\\ (x+7)^2=0\\$$$$$In the last stage we notice that the expression on the left side the term:

$(x+7)$

is multiplied by itself and therefore the expression can be written as a squared term:

$(x+7)^2$

Now that the expression on the left side has been factored into a product form (in this case not just a product but actually a power form) we will continue to the quick solution of the equation we received:

$(x+7)^2=0$

Let's pay attention to a simple fact, on the left side there is a term that is raised to the second power, and on the right side the number 0.

0 squared (to the second power) will give the result 0, so we get that the equation equivalent to this equation is the equation:

$x+7=0$(We could have solved algebraically and taken the square root of both sides of the equation, we'll discuss this in a note at the end)

We'll solve this equation by transferring the constant number to the other side and we'll get that the only solution is:

$x=-7$Let's summarize then the stages of solving the quadratic equation using the quick trinomial factoring method:

$x^2+14x+49=0 \\ \downarrow\\ (x+7)(x+7)=0\\ (x+7)^2=0\\ \downarrow\\ x+7=0\\ x=-7$Therefore, the correct answer is answer B.

Note:

We could have reached the final equation by taking the square root of both sides of the equation, however - taking a square root involves considering two possibilities: positive and negative (it's enough to consider this only on one side, as described in the calculation below), that is, we could have performed:

$(x+7)^2=0\hspace{8pt}\text{/}\sqrt{\hspace{8pt}}\\ \downarrow\\ \sqrt{(x+7)^2}=\pm\sqrt{0} \\ x+7=\pm0\\ x+7=0$

On the left side, the root (which is a half power) and the second power canceled each other out, and on the right side the root of 0 is 0, and we considered two possibilities positive and negative (this is the plus-minus sign indicated) except that the sign (which is actually multiplication by one or minus one) does not affect 0 which remains 0 in both cases, and therefore we reached the same equation we reached by logic - in the solution above.

In a case where on the right side there's a number other than 0, we could solve only by taking the root and considering the two positive and negative possibilities which would then give two different possibilities for the solution.

Let's solve the equation by simplifying the expression on the left side in two steps. First, we'll multiply the expressions in the two leftmost pairs of parentheses:

We'll use the shortened multiplication formula for squaring a binomial:

$(a\pm b)^2=a^2\pm2ab+b^2$

Since these two pairs of parentheses are being multiplied by another expression (which is also in parentheses), we'll put the result in parentheses (marked with an underline later):

$\underline{ (x-1)(x+1)}(x-2)=-2x^2+x^3 \\ \underline{ (x^2-1^2)}(x-2)=-2x^2+x^3 \\ (x^2-1)(x-2)=-2x^2+x^3$

Let's continue and simplify the expression on the left side using the expanded distribution law:

$(a+b)(c+d)=ac+ad+bc+bd$

Additionally, we'll use the law of exponents for multiplying terms with equal bases:

$a^ma^n=a^{m+n}$We'll now apply these laws and expand the parentheses in the expression in the equation:

$(x^2-1)(x-2)=-2x^2+x^3 \\ x^3-2x^2-x+2=-2x^2+x^3 \\$We'll continue and combine like terms, by moving terms between sides. Later - we can see that the terms with squared and cubed powers cancel out, therefore it's a first-degree equation, which we'll solve by isolating the variable term on one side and dividing both sides of the equation by its coefficient:

$x^3-2x^2-x+2=-2x^2+x^3\\ -x=-2\hspace{8pt}\text{/}:(-1)\\ \boxed{x=2}$

Therefore, the correct answer is answer C.

Let's solve the equation by simplifying the expression on the left side in two stages. First, we'll multiply the expressions within the two leftmost pairs of parentheses:

We'll use the shortened multiplication formula for squaring a binomial:

$(a\pm b)^2=a^2\pm2ab+b^2$

Since these two pairs of parentheses are being multiplied by another expression (which is also in parentheses), we'll put the result in parentheses (marked with an underline later):

$\underline{ (x-1)(x+1)}(x+1)=x^2+x^3 \\ \underline{ (x^2-1^2)}(x+1)=x^2+x^3 \\ (x^2-1)(x+1)=x^2+x^3$

Let's continue and simplify the expression on the left side using the expanded distribution law:

$(a+b)(c+d)=ac+ad+bc+bd$

Additionally, we'll use the law of exponents for multiplying terms with equal bases:

$a^ma^n=a^{m+n}$

We'll therefore apply these laws and expand the parentheses in the expression in the equation:

$(x^2-1)(x+1)=x^2+x^3 \\ x^3+x^2-x-1=x^2+x^3 \\$We'll continue and combine like terms, while moving terms between sides. Later - we can notice that the terms with squared and cubed powers cancel out, therefore it's a first-degree equation, which we'll solve by isolating the variable term and dividing both sides of the equation by its coefficient:

$x^3+x^2-x-1=x^2+x^3 \\ -x=1\hspace{8pt}\text{/}:(-1)\\ \boxed{x=-1}$

Therefore, the correct answer is answer A.

First, let's recall the formula for calculating the area of a rectangle:

The area of a rectangle (which has two pairs of equal opposite sides and all angles are $90\degree$) with sides of length $a,\hspace{4pt} b$units, is given by the formula:

$\boxed{ S_{\textcolor{blue}{\boxed{\hspace{6pt}}}}=a\cdot b }$(square units)

After recalling this fact, let's solve the problem:

Let's calculate the area of the rectangle whose vertices we'll mark with letters $EFGH$

$$It is given in the problem that one side of the rectangle is obtained by extending one side of the square with side length $x$ (cm) by 3 cm, and the second side of the rectangle is obtained by shortening the adjacent side of the given square by 3 cm:

$$

Therefore, the lengths of the rectangle's sides are:

$EF=HG=x+3\\ EH=FG=x-3$cm,

Now we'll use the above formula to calculate the area of the rectangle that was formed from the square in the way described in the problem:

$S_{\textcolor{blue}{\boxed{\hspace{6pt}}}}=EF\cdot EH\\ \downarrow\\ S_{\textcolor{blue}{\boxed{\hspace{6pt}}}}=(x+3)(x-3)$ (sq cm)

Let's continue and simplify the expression we got for the rectangle's area, using the difference of squares formula:

$(c+d)(c-d)=c^2-d^2$Therefore, we get that the area of the rectangle using the above formula is:

$S_{\textcolor{blue}{\boxed{\hspace{6pt}}}}=(x+3)(x-3) \\ S_{\textcolor{blue}{\boxed{\hspace{6pt}}}}=x^2-3^2\\ \boxed{ S_{\textcolor{blue}{\boxed{\hspace{6pt}}}}=x^2-9}$(sq cm)

Therefore, the correct answer is answer C.

First, let's recall the formula for calculating the area of a rectangle:

The area of a rectangle (which has two pairs of equal opposite sides and all angles are $90\degree$) with sides of length $a,\hspace{4pt} b$ units, is given by the formula:

$\boxed{ S_{\textcolor{blue}{\boxed{\hspace{6pt}}}}=a\cdot b }$(square units)

After recalling this fact, let's solve the problem:

Let's calculate the area of the rectangle whose vertices we'll mark with letters $EFGH$ (drawing)

$$It is given in the problem that one side of the rectangle is obtained by extending one side of the square with side length $x +1$ (cm) by 1 cm, and the second side of the rectangle is obtained by shortening the adjacent side of the given square by 1 cm:

Therefore, the lengths of the rectangle's sides are:

$EF=HG=(x+1)+1\\ \downarrow\\ \boxed{ EF=HG=x+2}\\ \hspace{2pt}\\ \\ EH=FG=(x+1)-1\\ \downarrow\\ \boxed{ EH=FG=x }$ (cm)

Now we'll use the above formula to calculate the area of the rectangle that was formed from the square in the way described in the problem:

$S_{\textcolor{blue}{\boxed{\hspace{6pt}}}}=EF\cdot EH\\ \downarrow\\ S_{\textcolor{blue}{\boxed{\hspace{6pt}}}}=(x+2)x$ (sq cm)

Let's continue and simplify the expression we got for the rectangle's area, using the distributive property:

$(m+n)d=md+nd$Therefore, using the distributive property, we get that the area of the rectangle is:

$S_{\textcolor{blue}{\boxed{\hspace{6pt}}}}=(x+2)x \\ \boxed{ S_{\textcolor{blue}{\boxed{\hspace{6pt}}}}=x^2+2x}$(sq cm)

Therefore, the correct answer is answer B.

Let's begin by focusing on the right side of the equation:

(2x-1)(2x+1)

We must first open the parentheses whilst multiplying all the terms as needed:

2x*2x+2x*1+-1*2x+-1*1

4x^2+2x-2x-1
4x^2-1

Let's now return to the original equation, and move all terms to the same side.

5X^2+7x+9=4X^2-1
5X^2-4X^2+7x+9+1=0
X^2+7X+10=0

We are left with a simple quadratic equation, which can be solved using any method we desire (factoring or the quadratic formula).

Therefore the final solution is:

X= -2,-5