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Slovenian Mathematical Olympiad, 2016

Algebra Difficulty 3.9 AMC 10/12 Find the answer Slovenia

Which quadratic equation has roots $1 - \sqrt{2}$ and $\frac{1}{\sqrt{2}-1}$ ?

Pick one

Solution

Rationalize the second root to get $\frac{1}{\sqrt{2}-1} = \frac{\sqrt{2}+1}{2-1} = \sqrt{2} + 1$ . These roots belong to the equation $(x - (1 - \sqrt{2})) (x - (\sqrt{2} + 1)) = 0$ . Expanding the left side we get $x^2 - 2x - 1 = 0$ . The correct answer is (D).

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