First slide

Theory of equations

Question

If roots of $a x^{2} + b x + c = 0, a, b, c \in R, a \neq 0$ are imaginary, then

Moderate

A

$a c > 0$
B

$a b > 0$
C

$b c > 0$
D

exactly two of $a b, b c$ and $c a$ are positive

Solution

Let $f (x) = a x^{2} + b x + c$ . If the roots of $f (x) = 0$
imaginary, then we cannot say anything about $b$ ( i.e. it can be
positive, negative or zero). So, options $(b), (c)$ and $(d)$ are not
necessarily true.
Further, if $a > 0,$ then the graph of $y = f (x)$ is above x-axis and hence
$f (x) > 0$ for all $x \in R \Rightarrow f (0) > 0 \Rightarrow c > 0$
$∴ a c > 0 .$
Similarly, if $a < 0,$ then the graph of $y = f (x)$ is below $x$ -axis and hence
$f (x) < 0$ for all $x \in R \Rightarrow f (0) < 0 \Rightarrow c < 0$
$∴ a c > 0$

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