TITLE: Roots of quadratic equation $ax^2+bx+c$ at infinity
QUESTION [5 upvotes]: Our maths teacher taught us that conditions for roots of a quadratic equation $ax^2+bx+c=0$ to lie at infinity are : 
A) (for exactly one root at infinity) $a=0$, $b$ = non-zero, & $c$ can be zero or non-zero 
B) (for both roots at infinity) $a = 0$, $b= 0$, and $c$ = non-zero 
By equal to zero, I mean the coefficients are so small that they are nearly zero, i.e approaching zero. They can’t be exactly equal to zero right? Or it would no longer be a quadratic. Please correct me if I’m wrong. 
This is not my question though. What I want to ask is, are these the only two conditions for roots of a quadratic equation to lie at infinity? 
For example, if we consider the equations 
1) $x^2-4\times 10^{30} = 0$
2) $0.00000001x^2+ 0.0000005x + 6 = 0$ 
I have attached two screenshots. In the first equation, both roots are at infinity even though coefficient of $x^2$ is non-zero (contrary to what condition (A) says. 
In the second equation, none of the roots are lying at infinity even thought coefficients of $x^2$ and $x$ are nearly equal to zero, which again is not consistent with what condition  (B) says. 
How do I explain these discrepancies? 
(Looks like I can’t attach more than one pic here. So I’m attaching just one, graph of equation (2). It’s clear that both roots of equation (1) are at infinity.)

REPLY [2 votes]: The roots to the quadratic equation $y = ax^2+bx+c$ are 
$$x_1
   =\dfrac{-b - \sqrt{b^2-4ac}}{2a}
   =\dfrac{2c}{-b + \sqrt{b^2-4ac}}$$
and 
$$x_2
   =\dfrac{-b + \sqrt{b^2-4ac}}{2a}
   =\dfrac{2c}{-b - \sqrt{b^2-4ac}}$$
If we let $a=0$ and $b > 0$, then the parabola "degenerates" into the line $y=bx+c$, whose root is 
$x = -\dfrac cb$.
If we look at $\left. \dfrac{-b - \sqrt{b^2-4ac}}{2a}\right|_{a=0}$ we get 
$x_1 =-\dfrac{-2b}{0}$; which we could interpret as $\infty$.
If we look at $\left. \dfrac{2c}{-b + \sqrt{b^2-4ac}}\right|_{a=0}$ we get 
$x_2 =-\dfrac cb$.
If we let $a=0$ and $b=0$, then the parabola degenerates into the constant $y=c$. 
If $c\ne 0$ then, in the right context, we might say that the roots are $\pm \infty$
If $c = 0$, then the roots are $(-\infty, \infty) = \mathbb R$.