What are the different ways to solve a quadratic equation?

There are three main ways to find the roots of a quadratic and they all have different benefits.Firstly, suppose your equation is in the form ax^2 + bx + c where a,b,c are all real numbers. Then you could find the two solutions (or the one repeated solution) by substituting the values a,b,c into the quadratic equation. However, you may have to find the square root of a negative number, in which case there are no real roots and the quadratic curve doesn't ever intersect with the x-axis. Is it easiest to show what happens by plotting the quadratic curve in this case, and seeing that it never crosses the x-axis.Secondly, it may be possible to find a factorization of the quadratic. This may be quicker than using the quadratic equation and if the factorization is of the form: (x-a)(x-b) then you know that the two roots are a and b. Once you find the two real roots (if they actually exist), then you could find the x-coordinate of the minimum point of this quadratic curve by simply finding the half-way point between the two roots. For example if x=2 and x=6 are your two roots, then the minimum point of the quadratic curve occurs at x=4.