Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app

JEE Main 2009

Exam: JEE Main
Year: 2009
Subject: Mathematics
Chapter: Quadratic Equation

JEE Main 2009 · Mathematics · Quadratic Equation

If the roots of the equation b x^2+c x+a=0 be imaginary, then for...

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

Options

Agreater than $4 a b$
Bless than $4 a b$
Cgreater than $-4 a b$
Dless than - 4ab

Correct answer

C. greater than $-4 a b$

Step-by-step solution

b x²+c x+a=0 Roots are imaginary

\Rightarrow c^2-4 a b \lt 0 \Rightarrow c^2 \lt 4 a b \Rightarrow-c^2 \gt -4 a b3 b^2 x^2+6 b c x+2 c^2

since

3 b^2 \gt 0

Given expression has minimum value Minimum value

=\frac{4\left(3 b^2\right)\left(2 c^2\right)-36 b^2 c^2}{4\left(3 b^2\right)}=-\frac{12 b^2 c^2}{12 b^2}=-c^2 \gt -4 a b

.

Practice Quadratic Equation on Quantrex Academy →

More from Quadratic Equation

Match each entry in List-I to the correct entry in List-II and choose the correct option.… JEE Advanced 2026 Let a, b, c be positive integers in arithmetic progression such that the equation ax^2 +… JEE Advanced 2026 Consider the quadratic equation (n^2 - 2n + 2)x^2 - 3x + (n^2 - 2n + 2)^2 = 0 , n ∈ ℝ .… JEE Main 2026 Let one root of the quadratic equation in x : (k^2 - 15k + 27)x^2 + 9(k-1)x + 18 = 0 be… JEE Main 2026 Let e_1 and e_2 be two distinct roots of the equation x^2 - ax + 2 = 0 . Let the sets \ a… JEE Main 2026 Let α, β be the roots of the equation x^2 - x + p = 0 and γ, δ be the roots the equation… JEE Main 2026 Let a, b ∈ ℂ . Let α, β be the roots of the equation x^2 + ax + b = 0 . If β - α = 11 and… JEE Main 2026 Let A, B , where A, B ∈ (-(π)/(2), (π)/(2)) , be the roots of the quadratic equation x^2… JEE Main 2026 If α=1 and β=1+i 2 , where i= -1 are two roots of the equation x^3+ax^2+bx+c=0 , a,b,c ∈… JEE Main 2026 If the quadratic equation (λ+2)x^2-3λ x+4λ=0 , λ ≠ -2 , has two positive roots, then the… JEE Main 2026 Let α, β be the roots of the equation x^2 - 3x + r = 0 , and (α)/(2), 2β be the roots of… JEE Main 2026 Let a, b, c ∈ \ 1, 2, 3, 4\ . If the probability, that ax^2 + 2 2 \,bx + c > 0 for all x… JEE Main 2026 Let S=\ x^ 3 +a x^ 2 +b x+c: a, b, c ∈ ~N. and .a, b, c ≤ 20\ be a set of polynomials.… JEE Main 2026 If α, β , where α<β , are the roots of the equation λ x^ 2 -(λ+3) x+3=0 such that… JEE Main 2026 The smallest positive integral value of a , for which all the roots of x^ 4 -a x^ 2 +9=0… JEE Main 2026 The sum of all the real solutions of the equation _ (x+3) (6 x^ 2 +28 x+30)=5-2 _ (6… JEE Main 2026 Full Quadratic Equation list Quadratic Equation JEE Maths PYQs JEE Mathematics PYQs All JEE Main PYQs