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:
$$ \color{blue}{ b = -14 } ~ \text{ and } ~ \color{red}{ c = 45 }$$Now we must discover two numbers that sum up to $ \color{blue}{ -14 } $ and multiply to $ \color{red}{ 45 } $.
Step 2: Find out pairs of numbers with a product of $\color{red}{ c = 45 }$.
PRODUCT = 45 | |
1 45 | -1 -45 |
3 15 | -3 -15 |
5 9 | -5 -9 |
Step 3: Find out which pair sums up to $\color{blue}{ b = -14 }$
PRODUCT = 45 and SUM = -14 | |
1 45 | -1 -45 |
3 15 | -3 -15 |
5 9 | -5 -9 |
Step 4: Put -5 and -9 into placeholders to get factored form.
$$ \begin{aligned} x^{2}-14x+45 & = (x + \color{orangered}{\square} )(x + \color{orangered}{\square}) \\ x^{2}-14x+45 & = (x -5)(x -9) \end{aligned} $$