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 and factorise.
`-20 = -20`
`16 - 36 = 25 - 45`
`4^2 - 4 * 9 = 5^2 - 5 * 9`
`(2 + 2)^2 - (2 + 2) * 9 = 5^2 - 5 * 9`
Add `81/4` to both sides.
`(2 + 2)^2 - (2 + 2) * 9 + 81/4 = 5^2 - 5 * 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) * 9 + 9^2/2^2 = 5^2 - 5 * 9 + 9^2/2^2`
`(2 + 2)^2 - (2 + 2) * 9 + (9/2)^2 = 5^2 - 5 * 9 + (9/2)^2`
`(2 + 2)^2 - 2 * (2 + 2) * 9/2 + (9/2)^2 = 5^2 - 2 * 5 * 9/2 + (9/2)`
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.
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?
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