If `a, b, c in R, a+b+c=0` then the quadratic equation `3ax^(2)+2bx+c=0` has __________ real roots.

Options

A. at least one root in (0, 1)
B. one root in (2, 3) and the other in (-2, -1)
C. imaginary roots
D. none of these

Answer

A

Explanation

Consider the function `phi(x) = ax^(3) + bx^(2) + cx`. Obviously `phi f(x)`, being a polynomial, is every where continous and differentiable
Also, `phi (0) = 0, phi (1) = a + b + c = 0" "["Given"]`
Therefore, by Rolle's Theorem there exists a point `lambda in (0, 1)` such that `phi'(lambda) = 0`.
We have, `phi'(x) = 3ax^(2) + 2bx + c`
`therefore" "phi'(lambda) = 0`
`rArr" "3a lambda^(2) + 2b lambda + c = 0`
`rArr" "lambda` is a root of `3ax^(2) + 2bx + c = 0`.
Thus, `3ax^(2) + 2bx + c = 0` has a root in (0, 1).