Quadratic Graphing Calculator — Vertex, Roots & Steps
Roots 2 and 3 · vertex (2.5, −0.25) · opens upward
Type standard form, for example 2x^2 - 3x - 5. Decimals and fractions such as 1/2x² are supported.
Solve ax² + bx + c = 0
Enter the three coefficients to find real or complex roots and analyze the parabola.
Interactive parabola graph
x: −1.5 to 6.5y: −1.53 to 17.03
Green points are real x-intercepts; gold marks the vertex and symmetry axis. The purple point and line show the selected function value and tangent.
Equation analysis
Two real roots. The parabola opens upward and its axis is x = 2.5. Domain: all real numbers; range: [-0.25, ∞).
x = (5 ± √1) / 2| Standard | x² − 5x + 6 = 0 |
| Vertex | (x − 2.5)² − 0.25 = 0 |
| Factored | (x − 2)(x − 3) = 0 |
| Axis | x = 2.5 |
| y-intercept | (0, 6) |
Solved graph examples
Step-by-step solution
Key points near the vertex
| Point | x | y = ax² + bx + c |
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Quadratic formulas and forms
Quadratic formula calculator
For ax² + bx + c = 0, the roots are x = (−b ± √(b² − 4ac)) / 2a. This solver calculates the discriminant first and returns two real roots, one repeated root, or a complex conjugate pair.
Vertex and axis of symmetry
The vertex lies at h = −b / 2a and k = f(h). The vertical line x = h is the axis of symmetry; the sign of a determines whether the parabola opens upward or downward.
Standard, vertex, and factored forms
Enter coefficients, start with a(x − h)² + k, or build an equation from two known roots. The calculator converts every input to standard form and shows the equivalent forms that exist.
Complex quadratic roots
When the discriminant is negative, the graph has no real x-intercepts, but the equation still has two complex roots. They are displayed as a conjugate pair using the imaginary unit i.
Display precision rounds the answers, not the entered coefficients. The graph automatically scales each axis; its on-screen angle is not the numerical slope. Calculations use floating-point arithmetic. The real-root evaluation follows the cancellation-resistant form described by Wolfram MathWorld.
Type a quadratic function to graph and solve it instantly, or edit standard, vertex and root forms. Trace the parabola and see roots, vertex, domain, range and steps.
