Step 1: Identify constants $ \color{blue}{ b }$ and $\color{red}{ c }$. ( $ \color{blue}{ b }$ is a number in front of the $ x $ term and $ \color{red}{ c } $ is a constant). In our case:
$$ \color{blue}{ b = 1 } ~ \text{ and } ~ \color{red}{ c = -72 }$$Now we must discover two numbers that sum up to $ \color{blue}{ 1 } $ and multiply to $ \color{red}{ -72 } $.
Step 2: Find out pairs of numbers with a product of $\color{red}{ c = -72 }$.
PRODUCT = -72 | |
-1 72 | 1 -72 |
-2 36 | 2 -36 |
-3 24 | 3 -24 |
-4 18 | 4 -18 |
-6 12 | 6 -12 |
-8 9 | 8 -9 |
Step 3: Find out which pair sums up to $\color{blue}{ b = 1 }$
PRODUCT = -72 and SUM = 1 | |
-1 72 | 1 -72 |
-2 36 | 2 -36 |
-3 24 | 3 -24 |
-4 18 | 4 -18 |
-6 12 | 6 -12 |
-8 9 | 8 -9 |
Step 4: Put -8 and 9 into placeholders to get factored form.
$$ \begin{aligned} x^{2}+x-72 & = (x + \color{orangered}{\square} )(x + \color{orangered}{\square}) \\ x^{2}+x-72 & = (x -8)(x + 9) \end{aligned} $$