# How do you solve and check for extraneous solutions in sqrt(23x+3)=14?

Aug 7, 2015

$x = \frac{193}{23}$ (no extraneous solutions)

#### Explanation:

Given $\sqrt{23 x + 3} = 14$

Square both sides:
$\textcolor{w h i t e}{\text{XXXX}}$$23 x + 3 = 196$

Subtract 3 from both sides
$\textcolor{w h i t e}{\text{XXXX}}$$23 x = 193$

Divide both sides by $23$
$\textcolor{w h i t e}{\text{XXXX}}$$x = \frac{193}{23}$$\textcolor{w h i t e}{\text{XXXX}}$approximately 8.391304 according to my calculator

Verify equation with $x = \frac{193}{23}$ to ensure this is not extraneous;
$\textcolor{w h i t e}{\text{XXXX}}$sqrt(23(193/23)+3)#

$\textcolor{w h i t e}{\text{XXXX}}$$= \sqrt{196}$

$\textcolor{w h i t e}{\text{XXXX}}$$= 14$

So $x = \frac{193}{23}$ is a valid result