I’ve been exploring quadratic equations and discovered a couple interesting patterns.

First is that b/a is equal to the width of the curve where it crosses the y-axis; this is also 2 times the real component of a complex solution.

The second is that the minimum height of a quadratic equation (for which there is only 1 possible x value) with a complex root is the square of the complex component multiplied by the leading coefficient.

Are these true features? Or am I just picking equations with favorable results? If so… well who cares? It’s fun to find these patterns, but I’m curious how they could be useful, or how I might take it a little bit further to find out more about the behavior of quadratics.