Question

If the roots of the equation $b{x}^2+cx+a=0$ is imaginary, then for all the real values of $x,$the expression $3b^2{x}^2+6bcx+2c^2$ is:

• A. Greater than $4ab$
• B. Less than $4ab$
• C. Greater than $-4ab$
• D. Less than $-4ab$

Answer 1

The correct option is C

Greater than $-4ab$

Explanation of correct answer.

Nature of roots:

Since the roots are imaginary, Therefore, the determinant will be negative.

The determinant of the equation $b{x}^2+cx+a=0$ is $c^2-4ab$.

Therefore,

$$\begin{gathered} c^2-4ab<0 \\ \Rightarrow c^2<4ab \end{gathered}$$

The minimum value of the expression is

$$\begin{gathered} -\frac{36b^2{c}^2-4\left(3b^2\right)\left(2c^2\right)}{4\left(3b^2\right)}=\frac{-12b^2{c}^2}{12b^2} \\ =-c^2 \end{gathered}$$

Now, we know,

$$\begin{gathered} c^2<4ab \\ \Rightarrow -c^2>-4ab \end{gathered}$$

Therefore, the correct option is (C)