Change the coefficients a, b, and c, and watch the function and the roots change.
The general quadratic expression is written as ax2 + bx + c. As you change the coefficients below, the graph redraws itself on the left. If the graph goes through whole-number points, they are highlighted.
The standard quadratic equation asks “When is the expression zero?” The roots, on the right, are plotted in the complex plane.
Watching the graphs change makes clear that there are always two roots: sometimes they are a pair of reals, sometimes a pair of complex conjugates.
In fact, this holds if you let the coefficients of x be complex numbers too; however the complex numbers need no longer be conjugates.