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# Difference of Squares

The difference of squares formula is one of the primary algebraic formulas used to expand a term in the form of $\left({a}^{2}-{b}^{2}\right)$ . Basically, it is an algebraic form of an expression used to equate the differences between two square values. The formula helps make a complex equation into a simple one.

## Formula to calculate the difference of squares

Any polynomial that can be written as ${a}^{2}-{b}^{2}$ can be factored as the difference of a square.

$\left({a}^{2}-{b}^{2}\right)=\left(a+b\right)\left(a-b\right)$

The reason is that when you use FOIL to expand the right side, the ab terms cancel out as follows:

$\left(a+b\right)\left(a-b\right)={a}^{2}-ab+ab-{b}^{2}={a}^{2}-{b}^{2}$

Example 1

Take the equation ${12}^{2}-{8}^{2}$

Using the difference of squares formula,

${a}^{2}-{b}^{2}=\left(a+b\right)\left(a-b\right)$

where

$a=12$

$b=8$

To calculate the left hand side (LHS),

${a}^{2}-{b}^{2}$

$={12}^{2}-{8}^{2}$

$=144-64$

$=80$

To calculate the right hand side (RHS),

$\left(a+b\right)\left(a-b\right)$

$=\left(12+8\right)\left(12-8\right)$

$=\left(20\right)\left(4\right)$

$=80$

$\mathrm{RHS}=\mathrm{LHS}$

This demonstrates that the formula works correctly.

Example 2

What is the value of $10{2}^{}-4{2}^{}$ ?

The formula for the difference of squares is:

${a}^{2}-{b}^{2}=\left(a+b\right)\left(a-b\right)$

Therefore, from the given expression,

$a=10$

$b=4$

${a}^{2}-{b}^{2}\to {10}^{2}-{4}^{2}$

$=\left(10+4\right)\left(10-4\right)$

$=\left(14\right)\left(6\right)$

$=84$

We can double check that this is correct by performing the calculations on the original problem.

${10}^{2}-{4}^{2}$

$=100-16$

$=84$

And we've simplified the problem correctly again.

## Factoring the difference of squares with variables

Frequently in algebra problems, the point of the difference of squares identity is to factor the statement, not to find the value of an expression as in the previous example. This can be done with variables, which you will usually find in algebraic problems.

Example 3

Factor, if possible.

${x}^{2}-49$

This is a difference of squares where $a=x$ and $b=7$

${x}^{2}-{7}^{2}=\left(x-7\right)\left(x+7\right)$

Example 4

Factor, if possible.

$16{p}^{2}-64{q}^{2}$

This is a difference of squares where $a=4p$ and $b=8q$ .

$16{p}^{2}-64{q}^{2}=\left(4p-8q\right)\left(4p+8q\right)$

We can be confident that these examples are factored correctly because of the first example where we solved the entire problem, making sure that the formula works properly.

## Interesting facts about difference of squares

• The difference of squares has a geometric interpretation. When you draw a square with sides of "a" length, you can "cut out" a square from one of the corners of that square with sides of "b" length. Then you can take the rectangle that's left from the "b" cutout, turn it 90 degrees, and connect it to the "a" rectangle. You now have a rectangle with side lengths $\left(a+b\right)$ and $\left(a-b\right)$ .
• The difference of squares is helpful in solving certain algebraic equations. One of these is the quadratic equation in the form ${x}^{2}-{c}^{2}=0$ , where ${x}^{2}$ and ${c}^{2}$ are perfect squares. Using the difference of squares, we can rewrite the equation as $\left(x+c\right)\left(x-c\right)=0$ , which allows us to find the solutions $x=c$ and $x=-c$ easily.
• The difference of squares has applications in number theory, as well. For example, it can help to prove that every odd integer can be expressed as the difference between two squares. For example, the odd integer 7 can be written as ${4}^{2}-{3}^{2}=16-9=7$ . Similarly, the odd integer 9 can be written as ${5}^{2}-{4}^{2}=25-16=9$ .

## Flashcards covering the Difference of Squares

Algebra 1 Flashcards

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