Multiple choice

If α and β are the roots of the equation x2 - 2x - 1 = 0, the value of (α2 + β2) is

  1. 8

  2. 12

  3. 6

  4. 2
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C Correct answer
Explanation

For x^2 - 2x - 1 = 0, sum of roots (a+b) = 2 and product (ab) = -1. Then a^2 + b^2 = (a+b)^2 - 2ab = 2^2 - 2(-1) = 4 + 2 = 6.

AI explanation

From the quadratic equation x^2 - 2x - 1 = 0, we use Vieta's formulas to find the sum of the roots (alpha + beta) is 2 and the product of the roots (alpha * beta) is -1. We use the algebraic identity for the sum of squares, alpha^2 + beta^2 = (alpha + beta)^2 - 2(alpha * beta). Substituting the known values gives (2)^2 - 2(-1) = 4 + 2 = 6.

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