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

Question

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 a b $, Less than $ 4 a b $, Greater than $ -4 a b $, Less than $ -4 a b $

MARKS App · Accepted Answer

Given b ⁡ x ⁡ 2 + c x + a = 0 has imaginary roots ⇒ c 2 - 4 a b 0 ⇒ c 2 4 a b ⇒ - c 2 > - 4 a b ...(i) Let f x = 3 b 2 x 2 + 6 b c x + 2 c 2 Here, 3 b 2 > 0 So, the given expression has a minimum value ∴ Minimum value = - D 4 a = 4 a c - b 2 4 a = 4 3 b 2 2 c 2 - 36 b 2 c 2 4 ( 3 b 2 ) = - 12 b 2 c 2 12 b 2 = - c 2 > - 4 a b [from Eq. (i)]

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