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Quadratic Formula Calculator

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 and green 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.

Graph

FAQ

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. The red dot shows root₁ = (5 + √(25−24))/2 = (5+1)/2 = 3. The green dot shows root₂ = (5−1)/2 = 2. 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.

Related Topics

Quadratic Equations (lesson)Factoring & RootsCubic Roots Solver