If $x\in R$ and the roots of $ax^2+bx+c=0$ are complex, then the sign of $a^2x^2+abx+ac$ is

Question

A) Always positive

B) Always negative

C)Always non-negative

D) Always zero

Obviously, this isn’t a solving question, so I can’t really show any working here (I would have otherwise). I didn’t really understand the question, so help would be greatly appreciated

Thanks!

Answer 1

The correct answer is always positive which is $(A)$

Note that $$a^2x^2+abx+ac=0 $$ is a parabola open upward and it has the same complex roots as the original equatin therefore it must be always positive.

Answer 2

HINT

$$ a^2x^2+abx+ac = a\left(ax^2+bx+c\right) $$

Answer 3

If the roots of $ax^2+bx+c=0$ are complex, the trinomial $ax^2+bx+c$ keeps a constant, non-zero sign in the reals, and so does $a^2x^2+abx+ac$. And for large enough $x$, this is perforce be positive.