Multiple choice

If $\alpha, \beta$ are the roots of the equations $x^{2} - 2x - 1 = 0$, then what is the value of $\alpha^{2} \beta^{-2} + \alpha^{-2}\beta^{2}$.

  1. $-2$
  2. $0$
  3. $30$
  4. $34$
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D Correct answer
Explanation

Given x^2 - 2x - 1 = 0, alpha+beta = 2, alpha*beta = -1. We want (alpha^2/beta^2) + (beta^2/alpha^2) = (alpha^4 + beta^4) / (alpha*beta)^2. alpha^2+beta^2 = (alpha+beta)^2 - 2*alpha*beta = 4 - 2(-1) = 6. alpha^4+beta^4 = (alpha^2+beta^2)^2 - 2(alpha*beta)^2 = 36 - 2(1) = 34. Result = 34 / (-1)^2 = 34.

AI explanation

By Vieta's formulas for x^2 - 2x - 1 = 0, we have alpha + beta = 2 and alpha*beta = -1. The expression can be rewritten using common denominators as (alpha/beta)^2 + (beta/alpha)^2, which equals ((alpha/beta) + (beta/alpha))^2 - 2. Simplifying the inner sum gives (alpha^2 + beta^2)/(alpha*beta), which equals ((alpha + beta)^2 - 2(alpha*beta))/(alpha*beta). Substituting the known values gives (2^2 - 2(-1)) / (-1) = -6. Plugging this back gives (-6)^2 - 2 = 36 - 2 = 34.

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