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?

Think about the graph of $y = ax^2 + bx + c$ (where $a \ne 0$). It's a parabola. It may be concave or convex, depending on whether $a>0$ or $a<0$, but it's still a parabola. The roots of $ax^2 + bx + c = 0$ are the places where $y=0$; in other words, the places where this parabola crosses the $x$-axis. Usually, there will be either two crossings, or none. There's a special case when the parabola just touches the axis. In this case, the quadratic has two equal roots. 
This isn't a "proof", but it may help your intuition.
