Question

The number of solution's of the equation $\sqrt{x+1}-\sqrt{x-1}=\sqrt{4x-1}$ is / are

• A. $2$
• B. $0$
• C. $3$
• D. $1$

Answer 1

The correct option is B

$0$

Find the number of solution for the given equation

Given:$\sqrt{x+1}-\sqrt{x-1}=\sqrt{4x-1}$

On squaring root on both sides, we get

$$\begin{gathered} \Rightarrow {\left(\sqrt{x+1}-\sqrt{x-1}\right)}^2={\left(\sqrt{4x-1}\right)}^2 \\ \Rightarrow \left(x+1\right)+(x-1)-2\sqrt{(x+1)(x-1)}=4x-1 \\ \Rightarrow 2x-2\sqrt{x^2-1}=4x-1 \\ \Rightarrow -2\sqrt{x^2-1}=2x-1 \\ \Rightarrow 4\left(x^2-1\right)={\left(2x-1\right)}^2 \\ \Rightarrow 4x^2-4=4x^2+1-4x \\ \Rightarrow 4x=5 \\ \Rightarrow x=\frac{5}{4} \end{gathered}$$

Substituting $x=\frac{5}{4}$ in given equation we get,

$$\begin{gathered} \sqrt{\frac{5}{4}+1}-\sqrt{\frac{5}{4}-1}=\sqrt{4\times \frac{5}{4}-1} \\ \Rightarrow \sqrt{\frac{9}{4}}-\sqrt{\frac{1}{4}}=\sqrt{5-1} \\ \Rightarrow \frac{3}{2}-\frac{1}{2}=2 \\ \Rightarrow 1\ne 2 \end{gathered}$$

$\therefore x=\frac{5}{4}$ not possible as it does not satisfy the given equation.

Therefor, no solution will exist for the given equation.

Hence, option (B) is the correct answer.