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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.

\begin{align*} &(a+b)^2 = a^2 + 2ab + b^2,\\ &(a-b)^2 = a^2 - 2ab + b^2,\\ &(a+b)(a-b) = a^2 - b^2. \end{align*}

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 \end{align*}