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
4^{2} – 4 x 9 ^{} = 5^{2} – 5 x 9
(2 + 2)^{2} – (2 + 2) x 9 = 5^{2} – 5 x 9
Add ^{81}/_{4} to both sides.
(2 + 2)^{2} – (2 + 2) x 9 ^{} + ^{81}/_{4} = 5^{2} – 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 ^{} + ^{9²}/_{2²} = 5^{2} – 5 x 9 + ^{9²}/_{2²}
(2 + 2)^{2} – (2 + 2) x 9 ^{} + (^{9}/_{2})^{2} = 5^{2} – 5 x 9 + (^{9}/_{2})^{2}
(2 + 2)^{2} – 2 x (2 + 2)^{} x ^{9}/_{2} + (^{9}/_{2})^{2} = 5^{2} – 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 a^{2} + 2ab + b^{2}.
((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.
Hint (highlight to read):
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?
