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If x = 3 - \(\sqrt{3}\), find x2 + \(\frac{36}{x^2}\)


Question

If x = 3 - \(\sqrt{3}\), find x2 + \(\frac{36}{x^2}\)

Options

A) 9

B) 18

C) 24

D) 27

The correct answer is C.

Explanation:

x = 3 - \(\sqrt{3}\)
x2 = (3 - \(\sqrt{3}\))2
= 9 + 3 - 6\(\sqrt{34}\)
= 12 - 6\(\sqrt{3}\)
= 6(2 - \(\sqrt{3}\))
∴ x2 + \(\frac{36}{x^2}\) = 6(2 - \(\sqrt{3}\)) + \(\frac{36}{6(2 - \sqrt{3})}\)
6(2 - \(\sqrt{3}\)) + \(\frac{6}{2 - \sqrt{3}}\) = 6(- \(\sqrt{3}\)) + \(\frac{6(2 + \sqrt{3})}{(2 - \sqrt{3})(2 + \sqrt{3})}\)
= 6(2 - \(\sqrt{3}\)) + \(\frac{6(2 + \sqrt{3})}{4 - 3}\)
6(2 - \(\sqrt{3}\)) + 6(2 + \(\sqrt{3}\)) = 12 + 12
= 24

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Dicussion (1)

  • x = 3 - \(\sqrt{3}\)
    x2 = (3 - \(\sqrt{3}\))2
    = 9 + 3 - 6\(\sqrt{34}\)
    = 12 - 6\(\sqrt{3}\)
    = 6(2 - \(\sqrt{3}\))
    ∴ x2 + \(\frac{36}{x^2}\) = 6(2 - \(\sqrt{3}\)) + \(\frac{36}{6(2 - \sqrt{3})}\)
    6(2 - \(\sqrt{3}\)) + \(\frac{6}{2 - \sqrt{3}}\) = 6(- \(\sqrt{3}\)) + \(\frac{6(2 + \sqrt{3})}{(2 - \sqrt{3})(2 + \sqrt{3})}\)
    = 6(2 - \(\sqrt{3}\)) + \(\frac{6(2 + \sqrt{3})}{4 - 3}\)
    6(2 - \(\sqrt{3}\)) + 6(2 + \(\sqrt{3}\)) = 12 + 12
    = 24

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