Question

Which of the following cannot be true for the given quadratic equation $ax^2+bx+c = 0 ; c\ne0$ if ratio of roots of this equation is equal to it's reciprocal?

Answer 1

Let $p, q$ be the roots of the given equation.

According to the given condition,
$\dfrac{p}{q} = \dfrac{q}{p}\\
\Rightarrow p^2 = q^2\\
\Rightarrow p = \pm q$

Case $1$:
$a>0$ and $p=q$, $b\ne0$ and $c$ is positive.


Case $2$:
$a>0$ and $p=-q$, $b=0$ and $c$ is negative.


Case $3$:
$a<0$ and $p=q$, $b\ne0$ and $c$ is negative.


Case $4$:
$a<0$ and $p=-q$, $b=0$ and $c$ is positive.