Question

If $a+b+c=0,$ then the quadratic equation $3a{x}^2+2bx+c=0$ has

• A. both imaginary
• B. at least one real root in (0, 1)
• C. one root in (2,3) other in (3, 6)
• D. none of these

Answer 1

Answer: (B)

Let a + b + c = 0
Consider function $f(x)=a{x}^3+b{x}^2+cx$ which is continuous and differentiable being a polynomial.
Now, f (0) = 0, f (1) = a + b + c = 0 (given)
$\therefore$ By rolle’s theorem a point $\alpha \in (0,1)$ such that f ' ($\alpha$) = 0
$\Rightarrow 3a{\alpha }^2+2b\alpha +c=0$
$\Rightarrow$ a is root of $3a{x}^2+2bx+c=0$