Quadratic Equation Calculator with Steps, Discriminant and Graph
Enter a, b and c of ax² + bx + c = 0. You get both roots in exact form (like (1 ± √3)/2), the discriminant, the vertex and a graph of the parabola.
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Roots
x = 3, 2
Two real roots- Discriminant D = b² − 4ac
- 1
- Vertex (h, k)
- (5/2, −1/4)
- x₁
- 3
- x₂
- 2
a > 0: the parabola opens upwards; its lowest point is at x = 5/2.
Show working
- x² − 5x + 6 = 0 (a = 1, b = −5, c = 6)
- D = b² − 4ac = (−5)² − 4·1·6 = 1
- x = (−b ± √D) / 2a = (5 ± √1) / 2
- x₁ = 3, x₂ = 2
- h = −b/2a = 5/2, k = c − b²/4a = −1/4
How to use
- Bring the equation to the form ax² + bx + c = 0.
- Enter a, b and c. Decimals and fractions-as-decimals are fine.
- Read the roots, then open “Show working” for the quadratic formula steps.
Good to know
Formula: x = (−b ± √D) / 2a with D = b² − 4ac (Sridharacharya’s formula in Indian textbooks). D > 0 gives two real roots, D = 0 one repeated root and D < 0 two complex roots p ± qi.
Exact answers: when a, b and c are whole numbers or simple decimals, roots are shown in simplest surd form, e.g. x² − 8 = 0 gives ±2√2, not just ±2.828427. Decimal values are listed underneath.
The vertex (h, k) with h = −b/2a and k = c − b²/4a is the turning point of the graph. The small graph marks the vertex and any real roots so you can see where the curve cuts the x-axis.
Questions
What is the quadratic formula?
x = (−b ± √(b² − 4ac)) / 2a. It solves every quadratic equation ax² + bx + c = 0 with a ≠ 0.
What does the discriminant tell me?
D = b² − 4ac. If D > 0 there are two different real roots, if D = 0 one repeated root, and if D < 0 no real roots (two complex roots).
How are complex roots written?
As p ± qi, where i = √−1. For x² + 2x + 5 = 0 the roots are −1 ± 2i.
Can I use this for NCERT Class 10 questions?
Yes. Use it to check your answers to the chapter on quadratic equations; the working follows the same formula.
Is my data stored or sent anywhere?
No. The calculation runs entirely in your browser; nothing you type leaves your device.