A binary operation * is defined on the set of real numbers, by \(a * b = \frac{a}{b} + \frac{b}{a}\). If \((\sqrt{x} + 1) * (\sqrt{x} – 1) = 4\), find the value of x.
- 6
- 5
- 4
-
3
The correct answer is: D
Explanation
\((\sqrt{x} + 1) * (\sqrt{x} - 1) = 4 \implies \frac{\sqrt{x} + 1}{\sqrt{x} - 1} + \frac{\sqrt{x} - 1}{\sqrt{x} + 1} = 4\)
\(\frac{(\sqrt{x} + 1)(\sqrt{x} + 1) + (\sqrt{x} - 1)(\sqrt{x} - 1)}{(\sqrt{x} - 1)(\sqrt{x} + 1)}\)
= \(\frac{x + 2\sqrt{x} + 1 + x - 2\sqrt{x} + 1}{x - 1} \implies \frac{2x + 2}{x - 1} = 4\)
\(2x + 2 = 4x - 4 \therefore 4x - 2x = 2x = 2 + 4= 6\)
\(x = 3\)