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Mathematics Question on Complex Numbers and Quadratic Equations

The equation x+1−x−1=4x−1\sqrt{x+1}-\sqrt{x-1}=\sqrt{4x-1} has

A

no solution

B

one solution

C

two solutions

D

more than two solutions

Answer

no solution

Explanation

Solution

Since, \hspace25mm\sqrt{x+1}-\sqrt{x-1}=\sqrt{4x-1}
⇒      (x+1)+(x−1)−2x2−1=4x−1\Rightarrow\, \, \, \, \, \, (x+1)+(x-1)-2\sqrt{x^2-1}=4x-1
\Rightarrow\hspace25mm1-2x=2\sqrt{x^2-1}
⇒      1+4x2−4x=4x2−4\Rightarrow\, \, \, \, \, \, 1+4x^2-4x=4x^2-4
\hspace25mm 4x=5 \Rightarrow x=\frac{5}{4}
But it does not satisfy the given equation.
Hence, no solution exists.