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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

1) greater than $4 a b$
2) less than $4 a b$
3) greater than $-4 a b$
4) less than $-4 a b$
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(3)

Explanation

$b x^{2}+c x+a=0$

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 $=\frac{4\left(3 b^{2}\right)\left(2 c^{2}\right)-36 b^{2} c^{2}}{4\left(3 b^{2}\right)}$

$=-\frac{12 b^{2} c^{2}}{12 b^{2}}=-c^{2}>-4 a b$
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