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

Question

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, 4 a b, > 4 a c, > - 4 a c, - 4 a b

MARKS App · Accepted Answer

Given that a x 2 + b x + c = 0 is having imaginary roots i.e., D = b 2 - 4 a c 0 . Let 3 a 2 x 2 + 6 a b x + 2 b 2 = f x Here coefficient of x 2 is positive so it will have minimum value. i.e., minimum value = 4 A C - B 2 4 A = 24 a 2 b 2 - 36 a 2 b 2 12 a 2 = - b 2 but we know that b 2 4 a c ⇒ - b 2 > - 4 a c . so minimum value is greater than - 4 a c .

Search any question & find its solution

Looking for more such questions to practice?