Multiple choice

If $\alpha, \beta$ are the root of quadratic equation $ax^2+bx+c=0$, then $\displaystyle \left ( a\alpha +b \right )^{-3}+\left ( a\beta +b \right )^{-3}=$

  1. $\displaystyle \frac{b^{2}-2abc}{a^{3}c^{3}}$
  2. $\displaystyle \frac{b^{3}-2abc}{a^{2}c^{2}}$
  3. $\displaystyle \frac{b^{3}-3abc}{a^{3}c^{3}}$

  4. None of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Roots are alpha, beta. a*alpha+b = -c/alpha (from quadratic properties). The expression becomes (-alpha/c)^3 + (-beta/c)^3 = -(alpha^3 + beta^3)/c^3. Using alpha^3 + beta^3 = (alpha+beta)^3 - 3*alpha*beta*(alpha+beta) and Vieta's formulas, this simplifies to the expression in option C.

AI explanation

Substitute alpha for x in the given quadratic to get a(alpha)^2 + b(alpha) + c = 0, which rearranges to a(alpha)^2 + b(alpha) = -c, or alpha(a*alpha + b) = -c. Dividing by -c gives a*alpha + b = -c/alpha, and similarly a*beta + b = -c/beta. We need to find the sum of (-c/alpha)^(-3) + (-c/beta)^(-3), which equals -1/c^3 multiplied by ((1/alpha)^3 + (1/beta)^3). Using the identity A^3 + B^3 = (A + B)^3 - 3AB(A + B) with A = 1/alpha and B = 1/beta, and knowing alpha + beta = -b/a and alpha*beta = c/a, the expression evaluates to b^3 - 3abc divided by a^3*c^3.

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