If the roots of the equation $ax^2-bx+c=0$ are $\alpha, \beta$, then the roots of the equation $b^2cx^2-ab^2x+a^3=0$ are
- $\displaystyle \frac {1}{\alpha^3+\alpha \beta},\displaystyle \frac {1}{\beta^3+\alpha \beta}$
- $\displaystyle \frac {1}{\alpha^2+\alpha \beta}, \displaystyle \frac {1}{\beta^2+\alpha \beta}$
- $\displaystyle \frac {1}{\alpha^4+\alpha \beta}, \displaystyle \frac {1}{\beta^4+\alpha \beta}$
-
none of these
Given the roots of ax^2 - bx + c = 0 are alpha and beta, the transformation of the equation leads to the roots 1/(alpha^2 + alpha*beta) and 1/(beta^2 + alpha*beta).
From the first equation, the sum of the roots is alpha + beta = b/a and the product is alpha * beta = c/a. Dividing the second equation by a^3 gives (b^2*c/a^3)x^2 - (a*b^2/a^3)x + 1 = 0, which simplifies to (b/a)^2*(c/a)x^2 - (b/a)x + 1 = 0. Substituting the sum and product yields (alpha + beta)^2 * (alpha * beta) * x^2 - (alpha + beta)x + 1 = 0, which expands to alpha*beta*(alpha^2 + 2*alpha*beta + beta^2)*x^2 - (alpha + beta)x + 1 = 0. This polynomial factors as (alpha^2 + alpha*beta)(beta^2 + alpha*beta)x^2 - (alpha + beta)x + 1 = 0, which further splits into [(alpha^2 + alpha*beta)x - 1][(beta^2 + alpha*beta)x - 1] = 0, giving the roots 1/(alpha^2 + alpha*beta) and 1/(beta^2 + alpha*beta).