information:classification, structural:expansion, linguistic:voice_change
Geometric significance

In terms of coordinate geometry, an axis-aligned parabola is a curve whose (x, y) {\displaystyle (x,y)} -coordinates are the graph of a second-degree polynomial, of the form y = a x 2 + b x + c {\displaystyle \textstyle y=ax^{2}+bx+c}, where a {\displaystyle a}, b {\displaystyle b}, and c {\displaystyle c} are real-valued constant coefficients with a ≠ 0 {\displaystyle a\neq 0}.

Geometrically, the quadratic formula defines the points (x, 0) {\displaystyle (x,0)} on the graph, where the parabola crosses the x {\displaystyle x} -axis. Furthermore, it can be separated into two terms,

x = − b ± b 2 − 4 a c 2 a = − b 2 a ± b 2 − 4 a c 2 a. {\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}=-{\frac {b}{2a}}\pm {\frac {\sqrt {b^{2}-4ac}}{2a}}.}

The first term describes the axis of symmetry, the line x = − b 2 a {\displaystyle x=-{\tfrac {b}{2a}}}. The second term, b 2 − 4 a c / 2 a {\displaystyle \textstyle {\sqrt {b^{2}-4ac}}{\big /}2a}, gives the distance the roots are away from the axis of symmetry.

If the parabola's vertex is on the x {\displaystyle x} -axis, then the corresponding equation has a single repeated root on the line of symmetry, and this distance term is zero; algebraically, the discriminant b 2 − 4 a c = 0 {\displaystyle \textstyle b^{2}-4ac=0}.

If the discriminant is positive, then the vertex is not on the x {\displaystyle x} -axis but the parabola opens in the direction of the x {\displaystyle x} -axis, crossing it twice, so the corresponding equation has two real roots. If the discriminant is negative, then the parabola opens in the opposite direction, never crossing the x {\displaystyle x} -axis, and the equation has no real roots; in this case the two complex-valued roots will be complex conjugates whose real part is the x {\displaystyle x} value of the axis of symmetry.