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

Answer 1

The correct option is B greater than $-4ab$

given $b{x}^2+cx+a=0$ has imaginary roots
so $\Delta <0$
$c^2-4ba<0$
For $3b^2{x}^2+6bcx+2c^2$

the minimum or maximum value of $3b{x}^2+6bcx+2c^2$ is $-\Delta /4a$
$\left(4\right(3b^2\left)\right(2c^2)-36b^2{c}^2)/12b^2$
$\Rightarrow$$-c^2$
as we know $-c^2>=-4ab$
so the minimum value is $-4ab$