# How do you solve 5x^2-10x=23 by completing the square?

May 21, 2018

$x = 1 \pm \sqrt{\frac{28}{5}}$

#### Explanation:

$\text{using the method of "color(blue)"completing the square}$

• " the coefficient of the "x^2" term must be 1"

$\text{factor out 5}$

$5 \left({x}^{2} - 2 x\right) = 23$

• "add to both sides "(1/2"coefficient of x-term")^2

$5 \left({x}^{2} + \left(- 1\right) x \textcolor{red}{+ 1}\right) = 23 \textcolor{red}{+ 5} \leftarrow \text{note}$

$\Rightarrow 5 {\left(x - 1\right)}^{2} = 28$

$\Rightarrow {\left(x - 1\right)}^{2} = \frac{28}{5}$

$\textcolor{b l u e}{\text{take the square root of both sides}}$

$\sqrt{{\left(x - 1\right)}^{2}} = \pm \sqrt{\frac{28}{5}} \leftarrow \textcolor{b l u e}{\text{note plus or minus}}$

$\Rightarrow x - 1 = \pm \sqrt{\frac{28}{5}}$

$\text{add 1 to both sides}$

$\Rightarrow x = 1 \pm \sqrt{\frac{28}{5}} \leftarrow \textcolor{red}{\text{exact solutions}}$