# How do you factor x^2 - 45x + 450?

Sep 26, 2016

${x}^{2} - 45 x + 450 = \left(x - 30\right) \left(x - 15\right)$

#### Explanation:

Note that $45 = 30 + 15$ and $450 = 30 \cdot 15$

Hence:

${x}^{2} - 45 x + 450 = \left(x - 30\right) \left(x - 15\right)$

Alternatively, you could find it by completing the square:

$4 \left({x}^{2} - 45 x + 450\right) = 4 {x}^{2} - 180 x + 1800$

$\textcolor{w h i t e}{4 \left({x}^{2} - 45 x + 450\right)} = {\left(2 x - 45\right)}^{2} - {45}^{2} + 1800$

$\textcolor{w h i t e}{4 \left({x}^{2} - 45 x + 450\right)} = {\left(2 x - 45\right)}^{2} - 2025 + 1800$

$\textcolor{w h i t e}{4 \left({x}^{2} - 45 x + 450\right)} = {\left(2 x - 45\right)}^{2} - 225$

$\textcolor{w h i t e}{4 \left({x}^{2} - 45 x + 450\right)} = {\left(2 x - 45\right)}^{2} - {15}^{2}$

$\textcolor{w h i t e}{4 \left({x}^{2} - 45 x + 450\right)} = \left(\left(2 x - 45\right) - 15\right) \left(\left(2 x - 45\right) + 15\right)$

$\textcolor{w h i t e}{4 \left({x}^{2} - 45 x + 450\right)} = \left(2 x - 60\right) \left(2 x - 30\right)$

$\textcolor{w h i t e}{4 \left({x}^{2} - 45 x + 450\right)} = 4 \left(x - 30\right) \left(x - 15\right)$

So dividing both ends by $4$ we find:

${x}^{2} - 45 x + 450 = \left(x - 30\right) \left(x - 15\right)$