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. greater than $4ab$
• B. less than $4ab$
• C. greater than $-4ab$
• D. less than $-4ab$

Answer 1

Answer: (C)

$b{x}^2+cx+a=0$
Roots are imaginary $\Rightarrow c^2-4ab<0\Rightarrow -c^2>-4ab$
$3b^2{x}^2+6bcx+2c^2$
since $3b^2>0$
Given expression has minimum value
Minimum value $=\frac{4\left(3b^2\right)\left(2c^2\right)-36b^2{c}^2}{4\left(3b^2\right)}=-\frac{12b^2{c}^2}{12b^2}=-c^2>-4ab.$