Multiple choice

If $\displaystyle \alpha ,\beta $ are the roots of the quadratic equation $\displaystyle { x }^{ 2 }-8x+p=0$, find the value of p if $\displaystyle { \alpha }^{ 2 }+{ \beta }^{ 2 }=40$.

  1. 8

  2. 10

  3. 12

  4. 6
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C Correct answer
Explanation

For x^2 - 8x + p = 0, sum of roots alpha + beta = 8, product alpha*beta = p. We know alpha^2 + beta^2 = (alpha+beta)^2 - 2(alpha*beta) = 40. Substituting: 8^2 - 2p = 40, so 64 - 2p = 40, 2p = 24, p = 12.

AI explanation

Using the sum of roots formula for x^2 - 8x + p = 0, we know alpha + beta = 8. The identity for the sum of squares states alpha^2 + beta^2 = (alpha + beta)^2 - 2(alpha)(beta). Substituting the known values gives 40 = 8^2 - 2p, which simplifies to 40 = 64 - 2p. Solving for p results in 2p = 24, so p = 12.

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