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. \[\begin{aligned} x^2 - 16 &= \begin{pmatrix} \sqrt{x^2} - \sqrt{16} \end{pmatrix} \begin{pmatrix} \sqrt{x^2} + \sqrt{16} \end{pmatrix} \\ & = \begin{pmatrix} x - 4 \end{pmatrix} \begin{pmatrix} x + 4 \end{pmatrix} \end{aligned}\]