Question

if x = 2-√5/ 2+√5 and y = 2+√5/ 2-√5, find the value of x²-y².

Answer 1

We know that,

(x + y)(x - y) = x² - y²

Therefore,

$x^2 - y^2 = (\frac{2 - \sqrt{5}}{2 + \sqrt{5}} + \frac{2 + \sqrt{5}}{2 - \sqrt{5}})(\frac{2 - \sqrt{5}}{2 + \sqrt{5}} - \frac{2 + \sqrt{5}}{2 - \sqrt{5}}) \\ \\ = (\frac{(2 - \sqrt{5})^2 + (2 + \sqrt{5})^2}{(2 + \sqrt{5})(2 - \sqrt{5})})(\frac{(2 - \sqrt{5})^2 - (2 + \sqrt{5})^2}{(2 + \sqrt{5})(2 - \sqrt{5})}) \\ \\ = (\frac{9 - 4\sqrt{5} + 9 + 4\sqrt{5}}{4 - 5})(\frac{9 - 4\sqrt{5} - 9 - 4\sqrt{5}}{4 - 5}) \\ \\ = \frac{18}{- 1} \times \frac{- 8\sqrt{5}}{- 1} = \frac{- 144\sqrt{5}}{1} = - 144\sqrt{5}$

- 144√5 is the answer.

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Answer 2

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