Question

Please help me solve this maths factorisation!​

Answer 1

This process is often called "completing the square". Recall that

`(a+b)^2 = a^2 + 2ab + b^2`

Let

`x^2 + 4x - 8 = (x + p)^2 + q`

Expand the right side to get

`x^2 + 4x - 8 = x^2 + 2px + p^2 + q`

Then

`\begin{cases}2p = 4 \\ p^2 + q = -8\end{cases} \implies \boxed{p = 2 \text{ and } q=-12}`

More generally, we can use this method rewrite a generic quadratic as

`ax^2 + bx + c = a \left(x + \frac b{2a}\right)^2 - (b^2)/(4a) + c`