Question

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

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

Answer 1

Answer: (C)

$bx^{2}+cx+a=0$
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 $= \frac{4\left(3b^{2}\right)\left(2c^{2}\right)-36b^{2}c^{2}}{4\left(3b^{2}\right)}=-\frac{12b^{2}c^{2}}{12b^{2}}=-c^{2} > -4ab.$