JEE Advance - Mathematics (1983 - No. 31)
If one root of the quadratic equation $$a{x^2} + bx + c = 0$$ is equal to the $$n$$-th power of the other, then show that
$$${\left( {a{c^n}} \right)^{{1 \over {n + 1}}}} + {\left( {{a^n}c} \right)^{{1 \over {n + 1}}}} + b = 0$$$
The statement is always true for any quadratic equation.
The given condition implies the relationship between the coefficients a, b, and c.
Let the roots be α and αn. Use the sum and product of roots relationships.
Substitute the expressions for the roots in terms of coefficients into the given equation.
The provided expression involving a, c, n, and b can be derived using Vieta's formulas.