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Enter the three coefficients of a quadratic equation and this calculator solves it with the quadratic formula, giving both roots — real or complex — along with the discriminant that tells you which kind to expect and the vertex of the parabola.
Quadratic Formula Calculator: step-by-step worked example
Solve 2x² − 4x − 6 = 0.
- a (coefficient of x²)
- 2
- b (coefficient of x)
- -4
- c (constant)
- -6
- 1
Discriminant
Δ = b² − 4ac
(-4)² − 4 × 2 × (-6) = 64
- 2
Square root of the discriminant
√Δ (complex roots if Δ < 0)
√64 = 8
- 3
First root
x₁ = (−b + √Δ) ÷ 2a
(4 + 8) ÷ 4 = 3
- 4
Second root
x₂ = (−b − √Δ) ÷ 2a
(4 − 8) ÷ 4 = -1
- 5
Vertex
V(−b ÷ 2a, c − b² ÷ 4a)
= (1, -8)
- y = ax² + bx + c:10 → 10
Good to know: Δ > 0: two real roots; Δ = 0: one repeated root; Δ < 0: no real roots (complex pair). Quick check with Vieta: x₁ + x₂ = −b/a and x₁ × x₂ = c/a.
Quadratic Formula Calculator — how it works
The quadratic formula
For ax² + bx + c = 0 (with a ≠ 0), the roots are:
x = (−b ± √(b² − 4ac)) ÷ 2a
The discriminant decides the roots
The part under the root, b² − 4ac, is the discriminant, and its sign tells you what to expect:
| Discriminant | Roots |
|---|---|
| Positive | Two distinct real roots |
| Zero | One repeated real root |
| Negative | Two complex conjugate roots |
The vertex
Every quadratic is a parabola with a turning point (vertex) at x = −b ÷ 2a, and the y-value there is the minimum (if a > 0) or maximum (if a < 0). The calculator returns both coordinates.
Worked example
Solve x² − 3x + 2 = 0. Here a = 1, b = −3, c = 2, so the discriminant is (−3)² − 4·1·2 = 9 − 8 = 1 — positive, so two real roots. Applying the formula: x = (3 ± √1) ÷ 2 = (3 ± 1) ÷ 2, giving x = 2 and x = 1. The vertex sits at x = 3 ÷ 2 = 1.5, y = −0.25.
Frequently asked questions
What is the quadratic formula?+
x = (−b ± √(b² − 4ac)) ÷ 2a. It solves any quadratic equation ax² + bx + c = 0 for its two roots. The ± gives the two solutions, and a must be non-zero for the equation to be quadratic.
What does the discriminant tell me?+
The discriminant, b² − 4ac, reveals the nature of the roots before you solve. Positive means two different real roots, zero means one repeated real root, and negative means two complex (conjugate) roots.
What are complex roots?+
When the discriminant is negative, the square root is of a negative number, so the roots involve the imaginary unit i (√−1). They come as a conjugate pair like −1 + 2i and −1 − 2i, and the parabola never crosses the x-axis.
What is the vertex of a parabola?+
The turning point, at x = −b ÷ 2a. It is the lowest point when a is positive (the parabola opens up) and the highest when a is negative. The tool gives both the x and y coordinates.
Why must a not equal zero?+
If a is 0 the x² term disappears and the equation becomes linear (bx + c = 0), not quadratic — the formula divides by 2a, which would be division by zero. The calculator requires a non-zero a.
About this page
- Publisher:
- Kalkoro — Teknolojig Bilişim Teknolojileri A.Ş., İstanbul
- Written by:
- Ahmet Tur — Editor (Math, date & time, unit conversion, physics & chemistry, everyday tools)
- Reviewed by:
- Kübra Yıldız — Editor
Parameters come from official sources; the last update date is shown at the bottom of the page.
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