Question

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Solve: $\sqrt{x}+3=\sqrt{x+5}$

Short answer

Ans: $\sqrt{x}+3=\sqrt{x+5}\Rightarrow NoSolutionforx\in \mathrm{ℝ}$

Step-by-step solution

Step 1. Given information.

given expression$\sqrt{x}+3=\sqrt{x+5}$

Step 2. Square both sides of the equation. 

$$\begin{gathered} {(\sqrt{x}+3)}^2={\left(\sqrt{x+5}\right)}^2 \\ \Rightarrow x+9+6\sqrt{x}=x+5 \\ \Rightarrow 6\sqrt{x}=-4 \end{gathered}$$

Step 3. Again, Square both sides of the equation and solve it.

$$\begin{gathered} {(6\sqrt{x})}^2={(-4)}^2\Rightarrow 36x=16 \\ \Rightarrow x=\frac{4}{9} \end{gathered}$$

Check the answer.

$$\begin{gathered} \sqrt{x}+3=\sqrt{x+5} \\ \Rightarrow \sqrt{\frac{4}{9}}+3=\sqrt{\frac{4}{9}+5} \\ \Rightarrow \frac{2}{3}+3=\sqrt{\frac{49}{9}} \\ \Rightarrow \frac{11}{3}\ne \frac{7}{3} \\ \Rightarrow LHS\ne RHS \end{gathered}$$

Verify solutions:$x=\frac{4}{9}$False.