1. Tardigrade
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  3. Mathematics
  4. Given a , b , c ∈ R. Statement-1:If c<0 b2-4 a c<0 then 4 a+c>2 b. Statement-2: If D <0, then ax 2+ bx + c takes either positive value only or negative value only where D is the discriminant of quadratic expression.

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Q. Given $a , b , c \in R$.
Statement-1:If $c<0 \& b^2-4 a c<0$ then $4 a+c>2 b$.
Statement-2 : If $D <0$, then $ax ^2+ bx + c$ takes either positive value only or negative value only where $D$ is the discriminant of quadratic expression.

Complex Numbers and Quadratic Equations

Solution:

Let $f(x)=a x^2+b x+c$
Now, $f (0)= c =$ negative ( given) & $D = b ^2-4 ac =$ negative ( given)
$\Rightarrow f ( x )$ takes negative value only $\forall x \in R ; \therefore f (-2)<0$
$\Rightarrow 4 a-2 b+c<0 ; \therefore 4 a+c<2 b \Rightarrow $ Option (D) is correct