Question

Factor the quadratic expression $5p^2-p-18$ completely.

Answer 1

Factor the quadratic polynomial $\mathbf{5}{\mathit{p}}^{\mathbf{2}}\mathbf{-}\mathit{p}\mathbf{-}\mathbf{18}$.

First, find the product of the quadratic term $5p^2$ and the constant term $-18$, $5p^2\times \left(-18\right)=-90p^2$. Through a hit and trial process, we have to find two factors of $-90p^2$ that add up to give the linear term $-p$. The two factors that satisfy these conditions are $-10p$ and $9p$, because their sum is equal to the linear term $-p$ and their product is equal to $-90p^2$. Rewrite the term $-p$ in the expression $5p^2-p-18$ as $-10p+9p$.

Then factor by grouping the terms:

$\begin{array}{rcl}5p^2-p-18& =& 5p^2-10p+9p-18\\ & =& \left(5p^2-10p\right)+\left(9p-18\right)\\ & =& 5p\left(p-2\right)+9\left(p-2\right)\\ & =& \left(p-2\right)\left(5p+9\right)\end{array}$

Thus, $\mathbf{5}{\mathit{p}}^{\mathbf{2}}\mathbf{-}\mathit{p}\mathbf{-}\mathbf{18}$ is completely factored to obtain $\mathbf{(}\mathit{p}\mathbf{-}\mathbf{2}\mathbf{)}\mathbf{(}\mathbf{5}\mathit{p}\mathbf{+}\mathbf{9}\mathbf{)}$.