# How do you solve sqrt(5m-16)=m-2?

Jul 24, 2015

Rewrite the equation as a quadratic and solve for $m$.

#### Explanation:

The first thing you need to do in order to solve this equation is get rid of the radical term by squaring both sides of the equation

${\left(\sqrt{5 m - 16}\right)}^{2} = {\left(m - 2\right)}^{2}$

$5 m - 16 = {m}^{2} - 4 m + 4$

Move all your terms to one side of the equation to get

${m}^{2} - 9 m + 20 = 0$

Use the quadratic formula to determine the two solutions for this equation

${x}_{1 , 2} = \frac{9 \pm \sqrt{81 - 4 \cdot 1 \cdot 20}}{2}$

${x}_{1 , 2} = \frac{9 \pm \sqrt{1}}{2} \implies \left\{\begin{matrix}{x}_{1} = \frac{9 + 1}{2} = 5 \\ {x}_{2} = \frac{9 - 1}{2} = 4\end{matrix}\right.$

Check both solutions to make sure that both are valid, i.e. you don't have an extraneous solution.

For ${x}_{1}$ you have

$\sqrt{5 \cdot 5 - 16} = 5 - 2$

$\sqrt{9} = 3 \to {x}_{1}$ is $\textcolor{g r e e n}{\text{valid}}$

For ${x}_{2}$ you have

$\sqrt{5 \cdot 4 - 16} = 4 - 2$

$\sqrt{4} = 2 \to {x}_{2}$ is $\textcolor{g r e e n}{\text{valid}}$