Multiple choice

Let α and β (α > β) be the roots of the equation x2 - 8x + q = 0. If α2 - β2 = 16, then what is the value of q?

  1. -15

  2. -10

  3. 10

  4. 15
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D Correct answer
Explanation

Roots alpha, beta satisfy alpha + beta = 8 and alpha * beta = q. We have alpha^2 - beta^2 = (alpha + beta)(alpha - beta) = 16. So 8(alpha - beta) = 16, which means alpha - beta = 2. Solving alpha + beta = 8 and alpha - beta = 2 gives alpha = 5, beta = 3. Then q = alpha * beta = 15.

AI explanation

Using Vieta's formulas for x^2 - 8x + q = 0, the sum of the roots alpha + beta equals 8. We are given the difference of their squares as alpha^2 - beta^2 = 16, which factors to (alpha + beta)(alpha - beta) = 16. Substituting the sum of the roots gives 8(alpha - beta) = 16, meaning alpha - beta = 2. Solving the system alpha + beta = 8 and alpha - beta = 2 yields alpha = 5 and beta = 3. The product of the roots is q = alpha*beta = 5 * 3 = 15.

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