Question

Factorise $x^2-2x-8$

Answer 1

We need to factories and find the factors of the given quadratic equation $x^2-2x-8$

Let us assume that the equation is in the form $a{x}^2+bx+c$

We know that if $\alpha$ and $\beta$ are the two root of the quadratic equation $a{x}^2+bx+c$, then $\alpha +\beta =-\frac{b}{a}$ and $\alpha \beta =\frac{c}{a}$.

In such case the quadratic equation can be expressed as $\left(x-\alpha \right)\left(x-\beta \right)$

Now comparing the above equation we get $\alpha +\beta =2$ and $\alpha \beta =-8$.

On solving this we get $\alpha =4$ and $\beta =-2$.

Thus, the above quadratic expression can be expressed as:

$$\begin{gathered} x^2-2x-8 \\ =x^2-4x+2x-8 \\ =x(x-4)+2(x-4) \\ =(x-4)(x+2) \end{gathered}$$

Thus, $\left(x-4\right)$ and $\left(x+2\right)$ are the factors of the polynomial $x^2-2x-8$.