Simplify \(\frac{\sqrt{1 + x} + \sqrt{x}}{\sqrt{1 + x} - \sqrt{x}}\)
A
-2x - 2\(\sqrt{x (1 + x)}\)
B
1 + 2x + 2\(\sqrt{x (1 + x)}\)
C
\(\sqrt{x (1 + x)}\)
D
1 + 2x - 2\(\sqrt{x (1 + x)}\)
Verified Correct Answer: B.
Step-by-Step Mathematical Solution
\(\frac{\sqrt{1 + x} + \sqrt{x}}{\sqrt{1 + x} - \sqrt{x}}\)
= \((\frac{\sqrt{1 + x} + \sqrt{x}}{\sqrt{1 + x} - \sqrt{x}}) (\frac{\sqrt{1 + x} + \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}})\)
= \(\frac{(1 + x) + \sqrt{x(1 + x)} + \sqrt{x(1 + x)} + x}{(1 + x) - x}\)
= \(\frac{1 + 2x + 2\sqrt{x(1 + x)}}{1}\)
= \(1 + 2x + 2\sqrt{x(1 + x)}\)
Source: Myschool Past Questions Archive (Record #31412)