Factor polynomial $$ x^{2}-15x+54 $$

$$ x^{2}-15x+54 = ( x-6 )( x-9 ) $$

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:

$$ \color{blue}{ b = -15 } ~ \text{ and } ~ \color{red}{ c = 54 }$$

Now we must discover two numbers that sum up to $ \color{blue}{ -15 } $ and multiply to $ \color{red}{ 54 } $.

Step 2: Find out pairs of numbers with a product of $\color{red}{ c = 54 }$.

PRODUCT = 54
1     54	-1     -54
2     27	-2     -27
3     18	-3     -18
6     9	-6     -9

Step 3: Find out which pair sums up to $\color{blue}{ b = -15 }$

PRODUCT = 54 and SUM = -15
1     54	-1     -54
2     27	-2     -27
3     18	-3     -18
6     9	-6     -9

Step 4: Put -6 and -9 into placeholders to get factored form.

$$ \begin{aligned} x^{2}-15x+54 & = (x + \color{orangered}{\square} )(x + \color{orangered}{\square}) \\ x^{2}-15x+54 & = (x -6)(x -9) \end{aligned} $$

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