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

Hint: 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
\[\begin{align}
  & \Rightarrow {{c}^{2}}<4ab \\
 & \Rightarrow -{{c}^{2}}>-4ab.......(1) \\
\end{align}\]
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
\[\begin{align}
  & \Rightarrow 36{{b}^{2}}{{c}^{2}}-24{{b}^{2}}{{c}^{2}}+12y\ge 0 \\
 & \Rightarrow 3{{c}^{2}}-2{{c}^{2}}+y\ge 0 \\
 & \Rightarrow {{c}^{2}}+y\ge 0 \\
\end{align}\]
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.