Multiple choice

If $\alpha & \beta$ are roots of quadratic equation ${ x }^{ 2 }+13x+8=0$ then the value of ${ \alpha }^{ 4 }+{ \beta }^{ 4 }=$

  1. $23281$
  2. $23218$
  3. $23128$
  4. $23182$
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A Correct answer
Explanation

For x^2 + 13x + 8 = 0, alpha + beta = -13 and alpha * beta = 8. We need alpha^4 + beta^4. First, alpha^2 + beta^2 = (alpha + beta)^2 - 2(alpha * beta) = (-13)^2 - 2(8) = 169 - 16 = 153. Then, alpha^4 + beta^4 = (alpha^2 + beta^2)^2 - 2(alpha * beta)^2 = 153^2 - 2(8^2) = 23409 - 128 = 23281.

AI explanation

By Vieta's formulas, the sum of the roots alpha plus beta equals -13, and the product alpha times beta equals 8. We use the algebraic identity alpha^4 + beta^4 = ((alpha + beta)^2 - 2*alpha*beta)^2 - 2*(alpha*beta)^2 to evaluate the expression. Substituting the known values yields ((-13)^2 - 2*8)^2 - 2*(8)^2, which simplifies to (169 - 16)^2 - 128. Calculating further, we get 153^2 - 128, resulting in 23409 - 128 equals 23281.

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