# Factorisation of quadratic expressions

A quadratic expression is an expression involving a variable and its square.

$$\Tred{t^2}-1$$, $$\Tred{x^2}+\Tred{x}-1$$, $$\Tred{y}-3$$ are quadratic expressions.

$$\Torange{t^3}-\Tred{t}$$ and $$\Torange{\sqrt{x}}+\Tred{x}-1$$ are not quadratic expressions.

**Factorisation and simplification of quadratic expressions** use the following formulae.

Here are applications of the rules:

$$$ x^2 - 1 = (x-1)(x+1),\quad (2x-3)^2 = 4x^2 -12x +9$$$The first rule can be used to compute squares of numbers $$\ge 10$$

\begin{align*} 12^2 = (\Tblue{10}+\Tgreen{2})^2 & = \Tblue{10}^2 + 2 \times \Tblue{10}\times \Tgreen{2} + \Tgreen{2}^2\\ & = 100 + 40 + 4 = 144\\ 18^2 = (\Tblue{20}-\Tgreen{2})^2 & = \Tblue{20}^2 - 2\times \Tblue{20}\times \Tgreen{2} + \Tgreen{2}^2\\ & = 400 - 80 + 4 = 324 \end{align*}