If the roots of the equation $b x^2+c x+a=0$ be imaginary, then for all real values of $x$, the expression…
If the roots of the equation $b x^2+c x+a=0$ be imaginary, then for all real values of $x$, the expression $3 b^2 x^2+6 b c x+2 c^2$ is
greater than $4 a b$
less than $4 a b$
greater than $-4 a b$
less than - 4ab
Solution
$
b x^2+c x+a=0
$
Roots are imaginary $\Rightarrow c^2-4 a b < 0 \Rightarrow c^2 < 4 a b \Rightarrow-c^2>-4 a b$ $3 b^2 x^2+6 b c x+2 c^2$
since $3 b^2>0$
Given expression has minimum value
Minimum value $=\frac{4\left(3 b^2\right)\left(2 c^2\right)-36 b^2 c^2}{4\left(3 b^2\right)}=-\frac{12 b^2 c^2}{12 b^2}=-c^2>-4 a b$