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

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less than 4ab

greater than –4ab

less than –4ab

greater than 4ab

answer is B.

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Detailed Solution

$\begin{array}{l} T h e r o o t s o f b x^{2} + c x + a = 0 a r e \\ i m a g i n a r y t h e D < 0 \\ \Rightarrow c^{2} - 4 a b < 0 \\ \Rightarrow c^{2} < 4 a b \\ N o w 3 b^{2} x^{2} + 6 b c x + 2 c^{2} = 3 {(b x + c)}^{2} - c^{2} \geq - c^{2} \Rightarrow \geq - 4 a b \end{array}$