Q. 341 - Intermediate Algebra

Question

In the following exercises, solve.
$\sqrt{x+7}-\sqrt{x-5}=2$

Step-by-Step Solution

Answer

The solution of the given equation is $x=9$.

Step 1. Given information.

The given equation is:

$\sqrt{x+7}-\sqrt{x-5}=2$

Step 2. Simplify the equation.

The given equation can be written as:

$\sqrt{x+7}=2+\sqrt{x-5}$

Take square on both sides.

$\begin{array}{rcl}(\sqrt{x+7}{)}^2& =& (2+\sqrt{x-5}{)}^2\\ x+7& =& 2^2+2\left(2\right)\left(\sqrt{x-5}\right)+(\sqrt{x-5}{)}^2\\ x+7& =& 4+4\sqrt{x-5}+x-5\\ x+7-4+5-x& =& 4\sqrt{x-5}\\ 8& =& 4\sqrt{x-5}\end{array}$

Step 3. Solve the equation.

Divide both sides by 4.

$\begin{array}{rcl}\frac{8}{4}& =& \frac{4\sqrt{x-5}}{4}\\ 2& =& \sqrt{x-5}\end{array}$

Square on both sides.

$\begin{array}{rcl}{2}^2& =& (\sqrt{x-5}{)}^2\\ 4& =& x-5\\ 4+5& =& x\\ 9& =& x\end{array}$

Step 4. Conclusion.

The solution of the given equation is $x=9$.