# Factorising Confusion

Factorising can be a very useful and powerful technique, but it's possible to make mistakes which can lead to confusion.

### 2 + 2 = 5

Start with a number, say –20, and make an equation by letting it equal itself, then rearrange each side.

–20 = –20

16 – 36 = 25 – 45

42 – 4 x 9 = 52 – 5 x 9

(2 + 2)2 – (2 + 2) x 9 = 52 – 5 x 9

(2 + 2)2 – (2 + 2) x 9 + 81/4 = 52 – 5 x 9 + 81/4

Each side is now a perfect square, so can be factorised. We'll take several steps to rearrange it so it's easier to recognise as a perfect square.

(2 + 2)2 – (2 + 2) x 9 + / = 52 – 5 x 9 + /

(2 + 2)2 – (2 + 2) x 9 + (9/2)2 = 52 – 5 x 9 + (9/2)2

(2 + 2)2 – 2 x (2 + 2) x 9/2 + (9/2)2 = 52 – 2 x 5 x 9/2 + (9/2)2

This is easier to see how it can factorise, because each side is in the form a2 + 2ab + b2.

((2 + 2) – 9/2)2 = (5 – 9/2)2

From there, we just need to take the square root of each side and cancel the common term.

(2 + 2) – 9/2 = 5 – 9/2

2 + 2 = 5

What went wrong? This error is not something that just happens in factorising. It's a good example of what can happen when we're not paying attention.

If you have trouble spotting the point at which the maths stopped making sense, try evaluating each line. What happens when we square a negative number?

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