# How do you factor 5n²-20n+5?

Mar 15, 2016

Separate out the common factor $5$ and complete the square to find:

$5 {n}^{2} - 20 n + 5 = 5 \left(n - 2 - \sqrt{3}\right) \left(n - 2 + \sqrt{3}\right)$

#### Explanation:

I will make use of the difference of squares identity:

${a}^{2} - {b}^{2} = \left(a - b\right) \left(a + b\right)$

with $a = \left(n - 2\right)$ and $b = \sqrt{3}$.

$\textcolor{w h i t e}{}$
First separate out the common scalar factor $5$, then complete the square as follows:

$5 {n}^{2} - 20 n + 5$

$= 5 \left({n}^{2} - 4 n + 1\right)$

$= 5 \left({n}^{2} - 4 n + 4 - 3\right)$

$= 5 \left({\left(n - 2\right)}^{2} - {\left(\sqrt{3}\right)}^{2}\right)$

$= 5 \left(\left(n - 2\right) - \sqrt{3}\right) \left(\left(n - 2\right) + \sqrt{3}\right)$

$= 5 \left(n - 2 - \sqrt{3}\right) \left(n - 2 + \sqrt{3}\right)$