If a + b + c = 0, then the quadratic equation 3ax2 + 2bx + c... | MATH - CONDITION FOR COMMON ROOTS, QUADRATIC EXPRESSIONS AND POSITION OF ROOTS | SolveeIt

If a + b + c = 0, then the quadratic equation 3ax² + 2bx + c = 0 has –

At least one root in [0, 1]

One root in [2, 3] and other is [–2, –1]

Imaginary roots

None of these

Answer

At least one root in $0, 1$

Explanation

Let (x) = ax³ + bx² + cx, x Ī [0, 1]. Since is a polynomial function, is differentiable on the whole real line and in particular on [0, 1].

Also, (0) = 0 and (1) = a + b + c = 0.

Thus, all the conditions in the hypothesis of the Rolle’s

theorem are satisfied by the Rolle’s theorem there exists at

least one a Ī (0, 1) such that ¢(a) = 0

But ¢(x) = 3ax² + 2bx + c and (1) = a + b + c = 0

Hence, 3ax² + 2bx + c = 0 has a root in [0, 1].

Question: If a + b + c = 0, then the quadratic equation 3a*x*<sup>2</sup> + 2b*x* + c = 0 has –...