Question

Let $a, b, c \in R$ with $a>0$ such that the equation $a x^2+b c x+b^3+c^3-4 a b c=0$ has non-real roots. Let $P(x)=a x^2+b x+c$ and $Q(x)=a x^2+c x+b$, then

• A. $P ( x )>0$ for all $x \in R$ and $Q ( x )<0$ for all $x \in R$.
• B. $P ( x )<0$ for all $x \in R$ and $Q ( x )>0$ for all $x \in R$.
• C. neither $P(x)>0$ for all $x \in R$ nor $Q(x)>0$ for all $x \in R$.
• D. exactly one of $P ( x )$ or $Q ( x )$ is positive for all real $x$.

Answer 1

Answer: (D)

$ D<0 $
$\Rightarrow b^2 c^2-4\left(b^3+c^3-4 a b c\right) a<0 $
$\Rightarrow\left(b^2 c^2-4 a b^3\right)+\left(16 a^2 b c-4 a c^3\right)<0$
$\Rightarrow b^2\left(c^2-4 a b\right)-4 a c\left(c^2-4 a b\right)<0$
$\Rightarrow\left(b^2-4 a c\right)\left(c^2-4 a b\right)<0$
$\Rightarrow D_{P(x)} \cdot D_{Q(x)}<0$
$\therefore$ Exactly one of $P ( x )$ or $Q ( x )$ is positive for all real $x$.