Given \(\dfrac{\sqrt{5+x} + \sqrt{5-x}}{\sqrt{5+x} - \sqrt{5-x}} = 3\), then what is the value of x?
3
Use the componendo–dividendo rule. Let \(a = \sqrt{5+x}\) and \(b = \sqrt{5-x}\).
Step 1 – Apply componendo–dividendo:
If \(\dfrac{a+b}{a-b} = \dfrac{3}{1}\), then \(\dfrac{a}{b} = \dfrac{3+1}{3-1} = \dfrac{4}{2} = 2\).
Step 2 – Square both sides:
\(\dfrac{a^2}{b^2} = \dfrac{5+x}{5-x} = 4\)
Step 3 – Solve for x:
\(5 + x = 4(5 - x)\)
\(5 + x = 20 - 4x\)
\(5x = 15 \implies x = 3\)
Hence \(x = 3\).
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