Question

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

Answer 1

`(x+ 13)(x - 13)`

Explanation:

This is a difference of two squares so it factors as `(a - b)(a +b)`, 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`

`= (x + 13)(x - 13)`, since `x • x = x^2` and `13 • -13 = -169`.

`x^2 - 169 = (x+ 13)(x - 13)`

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) `4x^2 - 81`

c) `x^2 + 25`

d) `x^4 - 16`

Hopefully this helps. Best of luck in the future!

Answer 2

`(x+ 13)(x - 13)`

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

`(a+b)(a-b)`

In our example, `a=x^2`, and `b=sqrt169`, or `b=13`. We can plug this into our difference of squares expansion equation to get

`(x+13)(x-13)`

Hope this helps!