### Types of Quadratic

A quadratic expression is functions of the form ...

$$a{x}^{2}+bx+c$$

where $a$, $b$ and $c$ are constants.

or, factorised ...

$$(px+q)(rx+s)$$which is a factorised version of the first, where $p$, $q$, $r$ and $s$ are constants.

or, completed square form ...

$$\alpha (x+\beta )+\gamma $$again another version of the first, where $\alpha $, $\beta $ and $\gamma $ are constants.

### Fully factorise the following:

(Hint: always look for a common factor first, if available, then check if other forms of factorisation are possible)

- ${x}^{2}+8x$
- ${x}^{2}+4x+4$
- ${x}^{2}+4x-12$
- ${x}^{2}+4x+5$
- ${x}^{4}-4{x}^{3}$
- ${x}^{2}-12x+35$
- ${z}^{2}-z-72$
- ${y}^{2}+2y-120$
- ${m}^{2}-144$
- ${p}^{3}+5{p}^{2}-36p$
- ${q}^{4}-16{q}^{2}+64$
- ${p}^{2}{q}^{2}-6pq+9$

### Challenge

How many integer values of $a$ are there for which ${x}^{2}+ax+100$ is factorisable? How about ${x}^{2}+ax-100$?