Multiple choice

If $\alpha$ and $\beta$ are the roots of the equation $x^2+px+q=0$, then the value of $\dfrac {\alpha^2}{\beta}+\dfrac {\beta^2}{\alpha}$ is

  1. $\dfrac {p(3q-p^2)}{q}$
  2. $\dfrac {q(3p-q^2)}{p}$
  3. $\dfrac {p(3q+p^2)}{q}$

  4. None of these
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A Correct answer
Explanation

Sum of roots = -p, Product = q. Expression = (a^3 + b^3) / (ab) = ((a+b)^3 - 3ab(a+b)) / (ab) = ((-p)^3 - 3q(-p)) / q = (-p^3 + 3pq) / q = p(3q - p^2) / q.

AI explanation

By Vieta's formulas, the sum and product of the roots give alpha + beta = -p and alpha * beta = q. Finding a common denominator for the expression yields (alpha^3 + beta^3) / (alpha * beta). Using the sum of cubes identity, the numerator becomes (alpha + beta)((alpha + beta)^2 - 3*alpha*beta), which expands to (-p)((-p)^2 - 3q). Simplifying this gives p(3q - p^2), and dividing by the product q results in p(3q - p^2) / q. The correct result is p(3q - p^2) / q.

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