Question

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

• A. $<4ab$
• B. $>-4ab$
• C. $=-4ab$
• D. $>4ab$

Answer 1

The correct option is B

$>-4ab$

Explanation for the correct option

Range of quadratic expression:

The given equation: $b{x}^2+cx+a=0$.

Compare the given equation with the general form of the quadratic equation $A{x}^2+Bx+C=0$.

Thus, $A=b,B=c,C=a$.

Thus, the discriminant of the equation can be given by, $D=B^2-4AC$.

$\Rightarrow D=c^2-4ba$.

As the roots of the equations are imaginary, thus $D<0$.

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

Now consider the expression $3b^2{x}^2+6bcx+2c^2$.

$3b^2{x}^2+6bcx+2c^2=3b^2{x}^2+6bcx+3c^2-c^2$

$$\begin{gathered} =3b^2{x}^2+6bcx+3c^2-c^2 \\ =3\left(b^2{x}^2+2bcx+c^2\right)-c^2 \end{gathered}$$

$\Rightarrow 3b^2{x}^2+6bcx+2c^2=3{\left(bx+c\right)}^2-c^2$

As ${\left(bx+c\right)}^2\ge 0$, thus $\left(3b^2{x}^2+6bcx+2c^2\right)>-c^2$.

$\Rightarrow \left(3b^2{x}^2+6bcx+2c^2\right)>-4ab\left[\because c^2<4ab\right]$.

Hence, option(B) is the correct option i.e. $>-4ab$