Quadratic Equation Solver
Solve any quadratic equation ax² + bx + c = 0. Find real or complex roots, discriminant, and vertex.
Examples
x² − 5x + 6 = 0
Two real roots (2 and 3)
- a (x² coefficient)
- 1
- b (x coefficient)
- -5
- c (constant)
- 6
Root x₁
3
Discriminant
1
Root x₂
2
Vertex X
2.5
Vertex Y
-0.25
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How It Works
Formula
Variables, symbols and units
- Coefficient of x² (must be non-zero)
- Coefficient of x
- Constant term
Calculation method explained
The discriminant b²−4ac is computed first. If non-negative, real roots are found with the quadratic formula. If negative, complex roots are expressed as a±bi. The vertex is at x = −b/(2a).
Examples
x² − 5x + 6 = 01 · -5 → 3
Two real roots (2 and 3)
- a (x² coefficient)
- 1
- b (x coefficient)
- -5
- c (constant)
- 6
- Root x₁
- 3
x² + 1 = 01 · 0 → 0 + 1i
Complex roots (±i)
- a (x² coefficient)
- 1
- b (x coefficient)
- 0
- c (constant)
- 1
- Root x₁
- 0 + 1i
2x² − 4x − 6 = 02 · -4 → 3
Roots of a wider parabola
- a (x² coefficient)
- 2
- b (x coefficient)
- -4
- c (constant)
- -6
- Root x₁
- 3
Frequently Asked Questions
What is a quadratic equation?
A quadratic equation is a polynomial equation of degree 2 in the form ax² + bx + c = 0, where a ≠ 0.
What is the discriminant?
The discriminant Δ = b² − 4ac determines the nature of the roots: Δ > 0 gives two distinct real roots, Δ = 0 gives one repeated root, Δ < 0 gives two complex conjugate roots.
What is the quadratic formula?
x = (−b ± √(b² − 4ac)) / (2a). It gives the solutions of any quadratic equation.
What are complex roots?
When the discriminant is negative, the roots contain imaginary numbers (i = √−1), expressed as a ± bi.
What is the vertex of a parabola?
The vertex is the highest or lowest point. Its x-coordinate is −b/(2a) and y-coordinate is f(−b/(2a)).