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Rationalize \(\frac{2 - \sqrt5}{3 - \sqrt5}\)


Question

Rationalize \(\frac{2 - \sqrt5}{3 - \sqrt5}\)

Options

A) \(\frac{1 - \sqrt5}{2}\)

B) \(\frac{1 - \sqrt5}{4}\)

C) \(\frac{ \sqrt5 - 1}{2}\)

D) \(\frac{1 + \sqrt5}{4}\)

The correct answer is B.

Explanation:

\(\frac{2 - \sqrt5}{3 - \sqrt5}\) x \(\frac{3 + \sqrt5}{3 + \sqrt5}\)
\(\frac{(2 - \sqrt5)(3 + \sqrt5)}{(3 - \sqrt5)(3 + \sqrt5)}\) = \(\frac{6 +2\sqrt5 - 3\sqrt5 - \sqrt25}{9 + 3\sqrt5 - 3\sqrt5 - \sqrt25}\)
= \(\frac{6 - \sqrt5 - 5}{9 - 5}\)
= \(\frac{1 - \sqrt5}{4}\)

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Dicussion (1)

  • \(\frac{2 - \sqrt5}{3 - \sqrt5}\) x \(\frac{3 + \sqrt5}{3 + \sqrt5}\)
    \(\frac{(2 - \sqrt5)(3 + \sqrt5)}{(3 - \sqrt5)(3 + \sqrt5)}\) = \(\frac{6 +2\sqrt5 - 3\sqrt5 - \sqrt25}{9 + 3\sqrt5 - 3\sqrt5 - \sqrt25}\)
    = \(\frac{6 - \sqrt5 - 5}{9 - 5}\)
    = \(\frac{1 - \sqrt5}{4}\)

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