Question

Solve each equation.

$\sqrt{x}+\sqrt{x+5}=5$

Short answer

The solution is $x=4$.

Step-by-step solution

Step 1. Given Information.

Given, to solve the equation $\sqrt{x}+\sqrt{x+5}=5$.

Step 2. Calculation.

Simplifying the equation:

$\begin{array}{l}\sqrt{x}+\sqrt{x+5}=5\\ {\left(\sqrt{x}+\sqrt{x+5}\right)}^2={\left(5\right)}^2\\ \left(x\right)+2\left(\sqrt{x}\right)\left(\sqrt{x+5}\right)+\left(x+5\right)=25\\ 2\sqrt{x\left(x+5\right)}=25-5-2x\\ 2\sqrt{x\left(x+5\right)}=20-2x\\ \sqrt{x\left(x+5\right)}=10-x\\ {\left(\sqrt{x\left(x+5\right)}\right)}^2={\left(10-x\right)}^2\\ x^2+5x=100-20x+x^2\\ 5x+20x=100\\ 25x=100\\ x=4\end{array}$

Checking if the value of x satisfies the equation:

$\begin{array}{l}\sqrt{x}+\sqrt{x+5}=5\\ \sqrt{4}+\sqrt{4+5}=5\\ \sqrt{4}+\sqrt{9}=5\\ 2+3=5\\ 5=5\end{array}$

Hence the value of x satisfies the equation.

Step 3. Conclusion.

Hence, the solution is x=4.