Functions
AI Assistant

Quadratic Equations

Graph, explore, and understand parabolas with an AI tutor

A quadratic equation has the standard form y = ax² + bx + c, where a, b, and c are constants and a ≠ 0. Its graph is a parabola — a symmetric U-shaped curve that opens upward when a > 0 and downward when a < 0.

The vertex is the highest or lowest point on the parabola, located at x = −b / (2a). The discriminant Δ = b² − 4ac tells you the number of real roots: two roots when Δ > 0, one repeated root when Δ = 0, and no real roots when Δ < 0.

Use the sliders below to change a, b, and c and watch how the parabola transforms. Ask the AI anything — try "What happens when a is negative?" or "Find the vertex."

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FAQ

What is the quadratic formula?
The quadratic formula solves ax² + bx + c = 0 for x: x = (−b ± √(b² − 4ac)) / 2a. It works for any quadratic equation, even when factoring is difficult.
How do I find the vertex of a parabola?
For the standard form y = ax² + bx + c, the vertex is at x = −b/(2a). Plug that x back into the equation to get y. In vertex form y = a(x − h)² + k, the vertex is simply (h, k).
What does the discriminant tell you?
The discriminant Δ = b² − 4ac reveals the nature of the roots. If Δ > 0, the parabola crosses the x-axis at two points. If Δ = 0, it touches the x-axis at exactly one point (the vertex). If Δ < 0, the parabola doesn't cross the x-axis — the equation has no real solutions.
What is vertex form and why is it useful?
Vertex form is y = a(x − h)² + k, where (h, k) is the vertex. It makes graphing easy: you can read the vertex directly, and a controls the width and direction of the parabola.

Related Topics

Slope of a LineSystems of EquationsFactoring & Roots