If the roots of the equation bx^2 + cx + a = 0 be imaginary, then for all real values of x , the expression 3b^2 x^2 + 6bcx + 2c^2 is: - Mathematics MCQ AIEEE 2009 2026 | ExamTest.live

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If the roots of the equation b x^{2} + c x + a = 0 be imaginary, then for all real values of x, the expression 3 b^{2} x^{2} + 6 b c x + 2 c^{2} is:

Solution & Step-by-step Explanation

Roots of b x^{2} + c x + a = 0 are imaginary ⟹ c^{2} - 4 ba < 0 ⟹ c^{2} < 4 ab .Let f (x) = 3 b^{2} x^{2} + 6 b c x + 2 c^{2} . This is a quadratic in x with A = 3 b^{2} > 0 .The minimum value occurs at x = - \frac{B}{2 A} = - \frac{6 b c}{6 b ^{2}} = - \frac{c}{b} . f_{min} = 3 b^{2} (- \frac{c}{b})^{2} + 6 b c (- \frac{c}{b}) + 2 c^{2} = 3 c^{2} - 6 c^{2} + 2 c^{2} = - c^{2} .Since c^{2} < 4 ab, then - c^{2} > - 4 ab .Thus, 3 b^{2} x^{2} + 6 b c x + 2 c^{2} \geq - c^{2} > - 4 ab .

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If the roots of the equation b x^{2} + c x + a = 0 be imaginary, then for all real values of x, the expression 3 b^{2} x^{2} + 6 b c x + 2 c^{2} is:

greater than 4 ab

less than 4 ab

greater than - 4 ab

less than - 4 ab

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