On the number of possible solutions for a quadratic equation. Solving a quadratic equation will yield two roots:
$$\frac{-\sqrt{b^2-4 a c}-b} {2 a}$$
and:
$$\frac{\sqrt{b^2-4a c}-b}{2 a}$$
And I've been taught to answer it like:
$$\frac{\pm\sqrt{b^2-4a c}- b}{2 a}$$
Why does it yields only two solutions? Aren't there infinite solutions for that? Is there a proof on the number of possible solutions for a quadratic equation?

One can see that there are only two solutions from the way it is solved. If $ax^2+bx+c=0$ with $a\neq 0$, it follows that $ax^2+bx=-c$, so that $x^2+\frac{b}{a}x=-\frac{c}{a}$. Trying to complete the square, one obtains $(x+\frac{b}{2a})^2=x^2+\frac{b}{a}x+(\frac{b}{2a})^2=(\frac{b}{2a})^2-\frac{c}{a}=\frac{b^2-4ac}{4a^2}$. Taking square roots, the result follows.
