Let r and c be real numbers. If r and -r are roots of 5x³+cx²-10x+9=0, then c equals

Question

UPSC Quizroom · Accepted Answer

Given: r and -r are roots of the cubic equation
5x^3 + c x^2 - 10x + 9 = 0

Since r and -r are roots, x^2 - r^2 is a factor. So write:
5x^3 + c x^2 - 10x + 9 = (x^2 - r^2)(5x + k)

for some real number k.

Expand the right-hand side:
(x^2 - r^2)(5x + k) = 5x^3 + k x^2 - 5 r^2 x - k r^2

Compare coefficients with the original equation:

From the x-term: -5 r^2 = -10 → r^2 = 2

From the constant term: -k r^2 = 9 → -k * 2 = 9 → k = -9/2

Since k is the coefficient of x^2, we have c = k = -9/2

Answer: -9/2

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