The quadratic equation is just a step further from the linear equation, we still restrict the problem to a single unknown variable (usually called 'x’) and add one term: x² (or x*x). The general form of such an equation is: $$\textbf{f(x) = ax² + bx + c } ,\: \: where\: \: a \neq 0$$

They will frequently turn up in many areas and very often make an appearance as part of the overall solution within most of the real world problems in the fields of physics, astronomy, engineering, computing, architecture...

Let us play with it online and we will quickly see that:
Changing a is changing the opening of the parabola.
Changing b is changing the slope of the parabola at x = 0.
Changing c is changing the offset of the parabola.

f(x) = x²

f(x) = 2x²

f(x) = 2x² + x

f(x) = 2x² + x - 1