# How do you solve sqrt(3x-5)=x-5 and check your solution?

Jul 28, 2017

The solution is $S = \left\{10\right\}$

#### Explanation:

The equation is

$\sqrt{3 x - 5} = x - 5$

The conditions are :

$3 x - 5 \ge 0$, $x \ge \frac{5}{3}$

Squaring the equation

${\left(\sqrt{3 x - 5}\right)}^{2} = {\left(x - 5\right)}^{2}$

$3 x - 5 = {x}^{2} - 10 x + 25$

${x}^{2} - 13 x + 30 = 0$

$\left(x + 3\right) \left(x - 10\right) = 0$

Therefore,

$x + 3 = 0$, $\implies$, $x = - 3$, this solution is not valid since $x \ge \frac{5}{3}$

$x - 10 = 0$, $\implies$, $x = 10$

Verification

$L H S = \sqrt{3 x - 5} = \sqrt{30 - 5} = \sqrt{25} = 5$

$R H S = x - 5 = 10 - 5 = 5$

$L H S = R H S$

$Q E D$