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. Greater than $-4ab$
• B. Less than $-4ab$
• C. Greater than $4ab$
• D. Less than $4ab$

Answer 1

The correct option is A Greater than $-4ab$

Given that the roots of the equation $b{x}^2+cx+a=0$ is imaginary.

So,

$c^2-4ab<0$

$\begin{array}{c}\Rightarrow -c^2>-4ab.............\left(1\right)\\ Let\\ 3b^2{x}^2+6bx+2c^2-y=0\\ Forrealvaluesofxwehave\\ {\left(6bc\right)}^2-4\left(3b^2\right).\left(2c^2-y\right)\ge 0\\ \Rightarrow 36b^2{c}^2-12.b^2\left(2c^2-y\right)\ge 0\\ \Rightarrow 3c^2-2c^2+y\ge 0\\ \Rightarrow c^2+y\ge 0\\ \Rightarrow y\ge -c^2.....\left(2\right)\\ comparing\left(1\right)and\left(2\right)\\ Wegety>\left(-4ab\right)\end{array}$