Multiple choice

A quadratic equation has roots p and q. If p + q = 10 and p2 + q2 = 68, what is the quadratic equation?

  1. x2 - 10x + 25 = 0

  2. x2 - 10x + 20 = 0

  3. x2 - 10x + 16 = 0

  4. x2 - 10x + 34 = 0

  5. x2 - 10x + 50 = 0
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C Correct answer
Explanation

The quadratic equation is x^2 - (sum of roots)x + (product of roots) = 0. Sum = 10. Product = pq. We know (p+q)^2 = p^2 + q^2 + 2pq. 100 = 68 + 2pq, so 2pq = 32, pq = 16. The equation is x^2 - 10x + 16 = 0.

AI explanation

Using the algebraic identity for the sum of squares, we have p squared plus q squared equals the sum squared minus twice the product. Substituting the given values gives 68 equals 100 minus twice the product, so twice the product is 32 and the product pq is 16. A quadratic equation is formed by x squared minus the sum of the roots times x plus the product of the roots, which becomes x squared minus 10x plus 16 equals zero.

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