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:
$$ \color{blue}{ b = 11 } ~ \text{ and } ~ \color{red}{ c = 24 }$$Now we must discover two numbers that sum up to $ \color{blue}{ 11 } $ and multiply to $ \color{red}{ 24 } $.
Step 2: Find out pairs of numbers with a product of $\color{red}{ c = 24 }$.
PRODUCT = 24 | |
1 24 | -1 -24 |
2 12 | -2 -12 |
3 8 | -3 -8 |
4 6 | -4 -6 |
Step 3: Find out which pair sums up to $\color{blue}{ b = 11 }$
PRODUCT = 24 and SUM = 11 | |
1 24 | -1 -24 |
2 12 | -2 -12 |
3 8 | -3 -8 |
4 6 | -4 -6 |
Step 4: Put 3 and 8 into placeholders to get factored form.
$$ \begin{aligned} x^{2}+11x+24 & = (x + \color{orangered}{\square} )(x + \color{orangered}{\square}) \\ x^{2}+11x+24 & = (x + 3)(x + 8) \end{aligned} $$