# How do you factor the expression x^2 - 169?

Jan 10, 2016

$\left(x + 13\right) \left(x - 13\right)$

#### Explanation:

This is a difference of two squares so it factors as $\left(a - b\right) \left(a + b\right)$, where a and b are the square roots of the original expression. See proofs below.

Warning: Differences of squares only works when there is a minus between the two terms, and doesn't work if it is positive. A sum of squares can't be factored with real numbers

${x}^{2} - 169$

$= \left(x + 13\right) \left(x - 13\right)$, since x • x = x^2 and 13 • -13 = -169.

${x}^{2} - 169 = \left(x + 13\right) \left(x - 13\right)$

Below are a few exercises to practice yourself. Watch out for the trick question(s) near the end!!:)

1. Factor each expression completely

a) ${x}^{2} - 49$

b) $4 {x}^{2} - 81$

c) ${x}^{2} + 25$

d) ${x}^{4} - 16$

Hopefully this helps. Best of luck in the future!

Jun 30, 2018

$\left(x + 13\right) \left(x - 13\right)$

#### Explanation:

What we have is a difference of squares, which has the form

${a}^{2} - {b}^{2}$, where $a$ and $b$ are perfect squares, which factor as

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

In our example, $a = {x}^{2}$, and $b = \sqrt{169}$, or $b = 13$. We can plug this into our difference of squares expansion equation to get

$\left(x + 13\right) \left(x - 13\right)$

Hope this helps!