# How do you complete the square to solve c^2 - 45c + 324 = 0?

Feb 26, 2017

$c = 36 \text{ }$ or $\text{ } c = 9$

#### Explanation:

The difference of squares identity can be written:

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

Complete the square, then use this with $a = 2 x - 45$ and $b = 27$ as follows:

$0 = 4 \left({c}^{2} - 45 c + 324\right)$

$\textcolor{w h i t e}{0} = 4 {c}^{2} - 180 c + 1296$

$\textcolor{w h i t e}{0} = {\left(2 c\right)}^{2} - 2 \left(2 c\right) \left(45\right) + 2025 - 729$

$\textcolor{w h i t e}{0} = {\left(2 c - 45\right)}^{2} - {27}^{2}$

$\textcolor{w h i t e}{0} = \left(\left(2 c - 45\right) - 27\right) \left(\left(2 c - 45\right) + 27\right)$

$\textcolor{w h i t e}{0} = \left(2 c - 72\right) \left(2 c - 18\right)$

$\textcolor{w h i t e}{0} = 4 \left(c - 36\right) \left(c - 9\right)$

So:

$c = 36 \text{ }$ or $\text{ } c = 9$