Multiple choice

If $\alpha$ and $\beta$ are the roots of the equation $x^2-4x+2=0$, find the value of $\alpha^2+\beta^2$.

  1. 8

  2. 10

  3. 12

  4. 14
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C Correct answer
Explanation

alpha + beta = 4, alpha * beta = 2. alpha^2 + beta^2 = (alpha + beta)^2 - 2(alpha * beta) = 4^2 - 2(2) = 16 - 4 = 12.

AI explanation

Use the relationship between roots and coefficients for the equation x^2 - 4x + 2 = 0 to find the sum of the roots, alpha + beta = 4, and the product of the roots, alpha * beta = 2. Apply the algebraic identity for the square of a sum, which states alpha^2 + beta^2 = (alpha + beta)^2 - 2(alpha * beta). Substitute the known values into the identity to get 4^2 - 2(2). Calculate the final value as 16 - 4 = 12.

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