Phase plane of coefficients

The phase plane of coefficients is a way to visualize how the coefficients of a quadratic trinomial $ax^2 + bx + c$ affect its properties, especially the nature of its roots. By plotting the coefficients (usually $b$ and $c$) on a coordinate plane for a fixed $a$, we can see regions where the quadratic has two real roots, one real root, or no real roots.

Important properties

• The discriminant $D = b^2 - 4ac$ determines the nature of the roots.

• The curve $b^2 - 4ac = 0$ divides the phase plane into regions with different root types.

• For a fixed $a$, the set of points $(b, c)$ where $b^2 - 4ac = 0$ forms a parabola in the $(b, c)$-plane.

• Above this parabola ($b^2 - 4ac > 0$), the quadratic has two real roots; on the parabola, one real root; below, no real roots.