TS EAMCET · Maths · Quadratic Equation

If the roots of the quadratic equation $a x^{2} + b x + c = 0$ are imaginary, then for all real values of $x$ , the minimum value of the expression $3 a^{2} x^{2} + 6 a b x + 2 b^{2}$ is

A $< 4 a b$
B $> 4 a c$
C $> - 4 a c$
D $< - 4 a b$

Verified Solution

Answer & Solution

Correct Answer

(C) $> - 4 a c$

Step-by-step Solution

Detailed explanation

Given that ax2+bx+c=0 is having imaginary roots i.e., D=b2-4ac<0. Let 3a2x2+6abx+2b2=fx Here coefficient of x2 is positive so it will have minimum value. i.e., minimum value =4AC-B24A= 24a2b2-36a2b212a2=-b2 but we know that b2<4ac⇒-b2 >-4ac. so…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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If the roots of the quadratic equation ax2+bx+c=0 are imaginary, then for all real values of x, the minimum value of the expression 3a2x2+6abx+2b2 is

Step-by-step Solution

If the roots of the quadratic equation $a x^{2} + b x + c = 0$ are imaginary, then for all real values of $x$ , the minimum value of the expression $3 a^{2} x^{2} + 6 a b x + 2 b^{2}$ is