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2016 Math Kangaroo

Problem 9

Problem 9 · 2016 Math Kangaroo Medium
Algebra & Patterns substitution

If \(x^2 - 4x + 2 = 0\), then \(x + \dfrac{2}{x}\) equals

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Answer: E — 4
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Hint 1 of 2
You never need the actual value of \(x\); reshape the equation toward \(x + \tfrac{2}{x}\).
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Hint 2 of 2
Divide the whole equation by \(x\).
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Approach: divide through by x
  1. From \(x^2 - 4x + 2 = 0\), regroup as \(x^2 + 2 = 4x\).
  2. Divide both sides by \(x\): \(x + \dfrac{2}{x} = 4\).
  3. So the value is 4 (E).
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