# Difference of Two Squares

The difference of two squares formula provides us with a formula

## Formula - Difference of Two Squares

$a^2-b^2 = \begin{pmatrix}a+b\end{pmatrix} \begin{pmatrix}a-b\end{pmatrix}$ $\begin{pmatrix}a+b\end{pmatrix} \begin{pmatrix}a-b\end{pmatrix} = a^2 - b^2$

## Example

Using the difference of two squares formula, distribute each of the following paretheses:

1. $$\begin{pmatrix}x+3 \end{pmatrix}\begin{pmatrix}x-3 \end{pmatrix}$$

2. $$\begin{pmatrix} x-4 \end{pmatrix}\begin{pmatrix} x+4 \end{pmatrix}$$

3. $$\begin{pmatrix}2x+5 \end{pmatrix}\begin{pmatrix}2x-5 \end{pmatrix}$$

4. $$\begin{pmatrix}3x+2y \end{pmatrix}\begin{pmatrix}3x-2y \end{pmatrix}$$

## Solution +

### Solution

1. The expansion of $$\begin{pmatrix} x + 3 \end{pmatrix} \begin{pmatrix} x - 3 \end{pmatrix}$$ is: \begin{aligned} \begin{pmatrix} x + 3 \end{pmatrix}\begin{pmatrix} x - 3 \end{pmatrix} & = x^2 - 3^2 \\ &= x^2 - 9 \end{aligned}
2. The expansion of $$\begin{pmatrix} x -4 \end{pmatrix} \begin{pmatrix} x +4 \end{pmatrix}$$ is: \begin{aligned} \begin{pmatrix} x -4 \end{pmatrix} \begin{pmatrix} x +4 \end{pmatrix} & = x^2 - 4^2 \\ & = x^2 - 16 \end{aligned}
3. The expansion of $$\begin{pmatrix} 2x +5 \end{pmatrix} \begin{pmatrix} 2x -5 \end{pmatrix}$$ is: \begin{aligned} \begin{pmatrix} 2x +5 \end{pmatrix} \begin{pmatrix} 2x -5 \end{pmatrix} & = \begin{pmatrix} 2x \end{pmatrix}^2 - 5^2 \\ & = 2^2x^2 - 5^2 \\ & = 4x^2-25 \end{aligned}
4. The expansion of $$\begin{pmatrix} 3x+2y \end{pmatrix} \begin{pmatrix} 3x - 2y \end{pmatrix}$$ is: \begin{aligned} \begin{pmatrix} 3x+2y \end{pmatrix} \begin{pmatrix} 3x - 2y \end{pmatrix} &= \begin{pmatrix} 3x \end{pmatrix}^2 - \begin{pmatrix}2y\end{pmatrix}^2 \\ & = 3^2x^2 - 2^2y^2 \\ & = 9x^2-4y^2 \end{aligned}

## Quick Calculations

The difference of two squares formula can be used to do quick calculations.
For instance, say we have to calculate $$98\times 102$$, without a calculator.
Noticing that each of the two numbers in this product are $$2$$ units away from $$100$$, we can write: $98\times 102 = \begin{pmatrix} 100 - 2 \end{pmatrix} \begin{pmatrix} 100 + 2 \end{pmatrix}$ We can then use the difference of two squares formula to write: \begin{aligned} 98\times 102 &= \begin{pmatrix} 100 - 2 \end{pmatrix} \begin{pmatrix} 100 + 2 \end{pmatrix} \\ &= 100^2 - 2^2 \\ &= 10000 - 4 \\ & = 9996 \end{aligned}

### Tutorial

This technique for quick calculations using the difference of two squares is illustrated in the following tutorial.

## Example

Use the difference of two squares formula to calculate each of the following without a calculator:

1. $$19\times 21$$

2. $$28\times 32$$

3. $$92\times 108$$

4. $$43\times 57$$

## Solution +

### Solution

1. We can calculate $$19 \times 21$$ as follows: \begin{aligned} 19\times 21 &= (20-1)(20+1) \\ &=20^2-1^2 \\ &=400 - 1 \\ & = 399 \end{aligned}
2. We can calculate $$28 \times 32$$ as follows: \begin{aligned} 28\times 32 &= (30-2)(30+2) \\ & = 30^2-2^2 \\ & = 900 - 4 \\ & = 896 \end{aligned}
3. We can calculate $$92 \times 108$$ as follows: \begin{aligned} 92\times 108 &= (100-8)(100+8) \\ & = 100^2 - 8^2\\ & = 10000 - 64 \\ & = 9936 \end{aligned}
4. We can calculate $$43 \times 57$$ as follows: \begin{aligned} 43\times 57 &= (50-7)(50+7) \\ & = 50^2-7^2 \\ & = 2500 - 49 \\ & = 2451 \end{aligned}

## Factoring with the Difference of Two Squares

The difference of two squares is often used to factorize expressions.
In particular quadratic equations looking like: $x^2-k^2 = 0$ the trick is to see that, using the difference of two squares formula, this can we written $\begin{pmatrix}x-k\end{pmatrix}\begin{pmatrix} x + k \end{pmatrix}$

## Example

Using the difference of two squares formula, write each of the following in factored form:

1. $$x^2-16$$

2. $$4p^2 - 25$$

3. $$x^2-9y^2$$

4. $$100m^2-49n^2$$

## Solution +

### Solution

Using the method described above, we factorize as follows:

1. \begin{aligned} x^2-16 &= \begin{pmatrix} \sqrt{x^2}- \sqrt{16} \end{pmatrix} \begin{pmatrix} \sqrt{x^2}+ \sqrt{16} \end{pmatrix} \\ & = \begin{pmatrix} x - 4 \end{pmatrix} \begin{pmatrix} x + 4 \end{pmatrix} \end{aligned}
2. \begin{aligned} 4p^2 - 25 & = \begin{pmatrix} \sqrt{4p^2} - \sqrt{25} \end{pmatrix} \begin{pmatrix} \sqrt{4p^2} + \sqrt{25} \end{pmatrix} \\ & = \begin{pmatrix} 2p - 5 \end{pmatrix} \begin{pmatrix} 2p + 5 \end{pmatrix}\end{aligned}
3. \begin{aligned} x^2 - 9y^2 &= \begin{pmatrix} \sqrt{x^2} - \sqrt{9y^2} \end{pmatrix} \begin{pmatrix} \sqrt{x^2} + \sqrt{9y^2} \end{pmatrix} \\ &= \begin{pmatrix}x - 3y \end{pmatrix} \begin{pmatrix} x + 3y \end{pmatrix}\end{aligned}
4. \begin{aligned} x^2 - 16 &= \begin{pmatrix} \sqrt{x^2} - \sqrt{16} \end{pmatrix} \begin{pmatrix} \sqrt{x^2} + \sqrt{16} \end{pmatrix} \\ & = \begin{pmatrix} x - 4 \end{pmatrix} \begin{pmatrix} x + 4 \end{pmatrix} \end{aligned}