# Is it possible to factor y=x^2+5x-36? If so, what are the factors?

May 1, 2018

$y = {x}^{2} + 5 x - 36$ can be factored as $\textcolor{red}{\text{(x+9)) * color(blue)(} \left(x - 5\right)}$

#### Explanation:

Consider the generalized factoring:
${\textcolor{w h i t e}{\text{XXX")color(red)(""(x+a))^2 * color(blue)(} \left(x + b\right)}}^{2} = {x}^{2} + \left(\textcolor{red}{a} + \textcolor{b l u e}{b}\right) x + \left(\textcolor{red}{a} \cdot \textcolor{b l u e}{b}\right)$

Applying the right side to the given ${x}^{2} \textcolor{g r e e n}{+ 5} x \textcolor{m a \ge n t a}{- 36}$
we can see that we need two values $: \textcolor{red}{a}$ and $\textcolor{b l u e}{b}$
such that
[1] their sum is $\textcolor{g r e e n}{+ 5}$, and
[2] their product is $\textcolor{m a \ge n t a}{- 36}$

Note that [2] implies that one of the numbers must be positive and the other negative (it's the only way you can get a negative product),
so we can think of the sum as being a difference of the magnitudes of the numbers (with the larger number being positive since $\textcolor{g r e e n}{+ 5}$ is positive.

Checking possible factors of $\textcolor{m a \ge n t a}{- 36}$ that meet the given requirements, we quickly find $\textcolor{red}{+ 9}$ and $\textcolor{b l u e}{- 5}$