Question

Factor the quadratic expression $x^2-7x-18$ completely.

Answer 1

Factor the quadratic polynomial ${\mathit{x}}^{\mathbf{2}}\mathbf{-}\mathbf{7}\mathit{x}\mathbf{-}\mathbf{18}$.

Find the product of the quadratic term $x^2$ and the constant term $-18$, $x^2\times \left(-18\right)=-18x^2$ .

Through a hit and trial process, we have to find two factors of $-18x^2$ that add up to give the linear term $-7x$.

The two factors that satisfy these conditions are $-9x$ and $2x$, because their sum is equal to the linear term $-7x$ and their product is equal to $-18x^2$.

Rewrite $-7x$ in the expression $x^2-7x-18$ as $-9x+2x$.

Then factorize by grouping the terms:

$\begin{array}{rcl}{x}^2-7x-18& =& x^2-9x+2x-18\\ & =& \left(x^2-9x\right)+\left(2x-18\right)\\ & =& x\left(x-9\right)+2\left(x-9\right)\\ & =& \left(x-9\right)\left(x+2\right)\end{array}$

Hence, $x^2-7x-18$ is completely factorized to obtain $\left(x-9\right)\left(x+2\right)$.