# How do you factor completely  a^2 - 10ab + 3b^2?

Jan 30, 2017

${a}^{2} - 10 a b + 3 {b}^{2} = \left(a - \left(5 + \sqrt{22}\right) b\right) \left(a - \left(5 - \sqrt{22}\right) b\right)$

#### Explanation:

To factor this, complete the square then use the difference of squares identity, which can be written:

${A}^{2} - {B}^{2} = \left(A - B\right) \left(A + B\right)$

We use this with $A = \left(a - 5 b\right)$ and $B = \sqrt{22} b$ as follows:

${a}^{2} - 10 a b + 3 {b}^{2} = {a}^{2} - 10 a b + 25 {b}^{2} - 22 {b}^{2}$

$\textcolor{w h i t e}{{a}^{2} - 10 a b + 3 {b}^{2}} = {a}^{2} - 2 a \left(5 b\right) + {\left(5 b\right)}^{2} - 22 {b}^{2}$

$\textcolor{w h i t e}{{a}^{2} - 10 a b + 3 {b}^{2}} = {\left(a - 5 b\right)}^{2} - {\left(\sqrt{22} b\right)}^{2}$

$\textcolor{w h i t e}{{a}^{2} - 10 a b + 3 {b}^{2}} = \left(\left(a - 5 b\right) - \sqrt{22} b\right) \left(\left(a - 5 b\right) + \sqrt{22} b\right)$

$\textcolor{w h i t e}{{a}^{2} - 10 a b + 3 {b}^{2}} = \left(a - \left(5 + \sqrt{22}\right) b\right) \left(a - \left(5 - \sqrt{22}\right) b\right)$