Contact The Learning Centre
# Quadratic equations

### Factorising a quadratic

- Expressions of the type \(x^2 + bx + c\) can be factorised by using the result \((x + e) (x + f) = x^2 + (e + f) x + ef\)

Thus \(b\) is the sum of two numbers \((b = e+f\)) and \(c\) is the product \((c = e\times f\))
- For example to factorise \(x^2 + 7x + 6\), we need two numbers whose sum is equal to \(7\) and product is equal to \(6\)

We select the appropriate numbers by guessing and checking. Solution is \(6\) and \(1\), giving the factors \(( x + 6)\) and \((x + 1)\)
- A similar process can be done when the coefficient of \(x^2\) is a number other than \(1\)
- For example factorise \(6x^2 + 17x - 3\): we need two numbers whose product is \(6\), i.e. \(6\) and \(1\) or \(3\) and \(2\). We need two other numbers whose product is \(-3\), i.e. \(3\) and \(-1\) or \(-3\) and \(1\). The correct combination of these give: \[6x^2 + 17x - 3=(6x - 1) (x + 3)\]