Graphic Quadratic Root Finder

Move sliders — watch roots and vertex update live

The quadratic formula solves ax² + bx + c = 0 for any values of a, b, and c:

x = (−b ± √(b² − 4ac)) / 2a

This calculator pre-draws the parabola and plots the two roots (red dots) and the vertex (purple dot) analytically. As you drag the a, b, c sliders, every point moves instantly — no solving needed.

When b² − 4ac < 0, the discriminant is negative and the roots become complex — the parabola no longer crosses the x-axis and the root dots disappear. Ask the AI to explain what's happening at any step.

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What is the quadratic formula?
The quadratic formula is x = (−b ± √(b²−4ac)) / 2a. It gives the exact roots of any quadratic equation ax² + bx + c = 0. The ± sign means there are two solutions: one with addition and one with subtraction.
What is the discriminant and why does it matter?
The discriminant is Δ = b² − 4ac. If Δ > 0, there are two real roots (the parabola crosses x-axis twice). If Δ = 0, there is one repeated root (the vertex touches the x-axis). If Δ < 0, the roots are complex — the parabola floats entirely above or below the x-axis.
How do I solve x² − 5x + 6 = 0 using this calculator?
Set a = 1, b = −5, c = 6. Root₁ = (5 + √(25−24))/2 = (5+1)/2 = 3. Root₂ = (5−1)/2 = 2. Both are shown as red dots. You can verify: (x−2)(x−3) = x²−5x+6 ✓
What does the vertex represent?
The vertex is the turning point of the parabola — the highest point if a < 0, or the lowest if a > 0. Its x-coordinate is always −b/2a (halfway between the two roots). The y-coordinate is the minimum or maximum value of the function.
Why do roots disappear when I drag the slider?
When the discriminant b² − 4ac becomes negative, the square root is imaginary and no real roots exist. The parabola has shifted entirely above or below the x-axis. The dots are hidden because the roots are complex numbers, not real points on the graph.
What can it graph?
It can plot explicit, implicit, and parametric functions, add points and geometry, and animate sliders on the same graph.
Can I use voice or a photo?
Yes. You can talk to the tutor, upload a worksheet or handwritten problem, and let the graph update from that input.
Will it explain the steps?
Yes. The AI explains what it is drawing and why, so you see the answer on the graph instead of getting only a final number.