Larger Theory for root formula Consider the quadratic equation:
$$ax^2 + bx + c = 0$$
and the linear equation:
$$bx + c = 0$$.
We note the solution of the linear equation is 
$$x = -\frac{c}{b}.$$
We note the solution of the quadratic equation is
$$\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$.
Suppose we take the limit as $a$ approaches 0 on the quadratic equation: ideally we should get the expression $-\frac{c}{b}$ left over, but this is clearly not the case:
This means that the quadratic formula does not generalize the linear formula, it is instead streamlined to only solve quadratic equations.
What would the general formula be? Basically what is the formula such that if $a$ is non-zero is equivalent to the quadratic equation and if $a = 0$ breaks down to the linear case?

Hint $\ $ First rationalize the numerator, then take the limit as $\rm\:a\to 0.$
$$\rm \frac{-b + \sqrt{b^2 - 4ac}}{2a}\ =\ \frac{2c}{-b -\sqrt{b^2 - 4ac}}\ \to\ -\frac{c}b\ \ \ as\ \ \ a\to 0 $$
Remark $\ $ The quadratic equation for $\rm\,\ z = 1/x\,\ $ is $\rm\,\ c\ z^2+ b\ z + a = 0\,\ $ hence
$$\rm z\ =\ \dfrac{1}{x}\ =\ \dfrac{-b \pm \sqrt{b^2-4\:a\:c}}{2\:c} $$ 
Inverting the above now yields the sought limit as $\rm\:a\to 0,\:$ so effectively removing the apparent singularity at $\rm\: a = 0\:.\ $ 
