Multiple choice

The roots of the equation x2 + ax + b = 0 are p and q, then the equation whose roots are p2q and pq2 will be:

  1. x2 + abx + b3 = 0

  2. x2 - abx + b3 = 0

  3. bx2 + x + a = 0

  4. x2 + ax + ab = 0
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A Correct answer
Explanation

Roots are p, q. p+q = -a, pq = b. New roots are p^2q and pq^2. Sum = pq(p+q) = b(-a) = -ab. Product = p^3q^3 = (pq)^3 = b^3. Equation is x^2 - (sum)x + product = 0 => x^2 - (-ab)x + b^3 = 0 => x^2 + abx + b^3 = 0.

AI explanation

Given the roots of x2 + ax + b = 0 are p and q, we know from Vieta's formulas that p + q = -a and pq = b. The new equation has roots p2q and pq2, so the sum of the new roots is p2q + pq2 = pq(p + q). Substituting the known values gives b(-a) = -ab, and the product of the new roots is (p2q)(pq2) = (pq)3 = b3. A quadratic equation with a sum of roots S and product P is x2 - Sx + P = 0, so substituting our values gives x2 + abx + b3 = 0.

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