# Why does the difference of two squares work?

Dear Uncle Colin,

Why does the difference of two squares work?

- Doubtful Of Truth Status

Hi, DOTS, and thanks for your message! I have several ways of convincing yourself that $(a+b)(a-b)$ is the same thing as $a^2 - b^2$.

### Just multiply it out

This is the “obvious” way: if you multiply it out term by term, you get $a^2 + ab - ab - b^2$, and the $ab$s in the middle add up to nothing, leaving you with $a^2 - b^2$.

### A quadratic example

The next example works with $a$s and $b$s, but it’s a bit more obvious if I start with something like $x^2 - 25$, which is the sort of quadratic you’ve probably seen before.

If you write it as $x^2 + 0x - 25$, it’s even more familiar: if you want to factorise it, you need two numbers with a product of -25 and a sum of 0. You don’t have that many options: you want -5 and 5, so your brackets are $(x-5)(x+5)$.

### More quadratic

Another way to see it is to use the factor theorem: $(x-a)$ is a factor of $f(x)$ ((for a polynomial)) if and only if $f(a)=0$.

We’re looking at $f(a) = a^2 - 25$, which is equal to 0 when $a^2 = 25$. That gives $a = 5$ and $a = -5$, so the factors are $(x-5)$ and $(x+5)$.

Hope that helps!

- Uncle Colin