Question

If the roots of the equation $b{{x}^{2}}+cx+a=0$ be imaginary, then for all real values of x, the expression $3{{b}^{2}}{{x}^{2}}+6bcx+2{{c}^{2}}$ is?
A. Greater than 4ab
B. Less than 4ab
C. Greater than -4ab
D. Less than -4ab

Answer 1

In this problem, we are given that the equation $b{{x}^{2}}+cx+a=0$ has imaginary roots and we have to find value of the equation $3{{b}^{2}}{{x}^{2}}+6bcx+2{{c}^{2}}$ for all real values of x. Here we can use the discriminant formula where the discriminant value less than 0 has imaginary roots and the discriminant greater than or equal to zero has real roots.

Complete step by step solution:
We are given that the equation $b{{x}^{2}}+cx+a=0$ has imaginary roots.
We know that the discriminant value is less than zero for imaginary roots, we can write it as,
$\Rightarrow {{c}^{2}}-4ab<0$
We can now write the above step as

& \Rightarrow {{c}^{2}}<4ab \\\ & \Rightarrow -{{c}^{2}}>-4ab.......(1) \\\ \end{aligned}$$ We can now write another equation $$3{{b}^{2}}{{x}^{2}}+6bcx+2{{c}^{2}}$$. Let $$3{{b}^{2}}{{x}^{2}}+6bcx+2{{c}^{2}}=y$$ We can now write it as $$\Rightarrow 3{{b}^{2}}{{x}^{2}}+6bcx+2{{c}^{2}}-y$$ We know that the discriminant value is greater than or equal to zero for all real roots. $$\Rightarrow 36{{b}^{2}}{{c}^{2}}-4\left( 3{{b}^{2}} \right)\left( 2{{c}^{2}}-y \right)\ge 0$$ We can now simplify the above step, we get $$\begin{aligned} & \Rightarrow 36{{b}^{2}}{{c}^{2}}-24{{b}^{2}}{{c}^{2}}+12y\ge 0 \\\ & \Rightarrow 3{{c}^{2}}-2{{c}^{2}}+y\ge 0 \\\ & \Rightarrow {{c}^{2}}+y\ge 0 \\\ \end{aligned}$$ We can now write the above step as, $$\Rightarrow y\ge -{{c}^{2}}$$ …….. (2) We can now compare (1) and (2), we get $$\Rightarrow y>-4ab$$ **Therefore, the answer is option C. Greater than -4ab.** **Note:** We should always remember that for imaginary roots the discriminant value will be less than 0 and for real roots the discriminant value will be greater than or equal to zero. We should also remember that discriminant formula for the equation $$a{{x}^{2}}+bx+c=0$$ is $${{b}^{2}}-4ac$$ according to the equation, we have to change the variables.