# How do you solve the quadratic equation by completing the square: x^2-6x=0?

Jul 13, 2015

${x}^{2} - 6 = 0$
$\Rightarrow x = 0$ or $x = 6$
$\textcolor{w h i t e}{\text{XXXX}}$(by completing the square)

#### Explanation:

Given ${x}^{2} - 6 x = 0$

$\textcolor{w h i t e}{\text{XXXX}}$If ${x}^{2}$ and $- 6 x$ are the first two terms of a squared binomial:
$\textcolor{w h i t e}{\text{XXXX}}$$\textcolor{w h i t e}{\text{XXXX}}$${\left(x - a\right)}^{2} = \left({x}^{2} - 2 a x + {a}^{2}\right)$
$\textcolor{w h i t e}{\text{XXXX}}$Then (since $- 2 a x = - 6 a x$) $\Rightarrow a = 3$

$\textcolor{w h i t e}{\text{XXXX}}$and the third term must be ${a}^{2} = 9$

$\textcolor{w h i t e}{\text{XXXX}}$So we need to add $9$ (to both sides) to "complete the square"
${x}^{2} - 6 x + 9 = 9$

$\textcolor{w h i t e}{\text{XXXX}}$Rewriting as a squared binomial
${\left(x - 3\right)}^{2} = 9$

$\textcolor{w h i t e}{\text{XXXX}}$Taking the square root of both sides
$x - 3 = \pm \sqrt{9} = \pm 3$

$\textcolor{w h i t e}{\text{XXXX}}$Adding 3 to both sides
$x = 6$ or $x = 0$

(Note that, in this case, the solution would be simpler to determine by factoring).