Quadratic Equation Solver

Solve any quadratic equation ax² + bx + c = 0 instantly. Find real and complex roots, vertex, axis of symmetry, discriminant, and see the parabola plotted on a live chart.

A quadratic equation calculator solves equations of the form ax² + bx + c = 0, finding the values of x (roots) where the parabola crosses the x-axis. Quadratic equations appear throughout physics, engineering, economics, and geometry. The discriminant (b² − 4ac) determines the nature of the roots: positive = two real roots; zero = one repeated root; negative = two complex roots. Prime Factorization Calculator and Triangle Solver Calculator provide related mathematical tools.

Enter coefficients a, b, and c from your equation ax² + bx + c = 0.
The calculator computes the discriminant and applies the quadratic formula.
Read the roots x₁ and x₂ — these are the x-values where the quadratic equals zero.
If the discriminant is negative, the roots are complex (no real solutions).

Quadratic formula

x = (−b ± √(b² − 4ac)) ÷ 2a

Discriminant Δ = b² − 4ac: Δ > 0 → two distinct real roots; Δ = 0 → one repeated root; Δ < 0 → two complex roots.

Example: x² − 5x + 6 = 0. a=1, b=−5, c=6. Δ = 25−24 = 1. x = (5 ± 1) ÷ 2. Roots: x₁ = 3, x₂ = 2.

Interpreting roots

Types of solutions

Two real roots mean the parabola crosses the x-axis twice. One repeated root means it touches but doesn't cross. Complex roots mean the parabola doesn't cross the x-axis — the equation has no real solutions. Vertex of the parabola: x = −b÷(2a), y = c − b²÷(4a).

Math tips and best practices

Always check a ≠ 0 — if a = 0 the equation is linear, not quadratic.
Factor first if possible — if the roots are integers, factoring is faster than the formula.
The sum of roots = −b/a; the product of roots = c/a — useful checks.

The quadratic formula was known to Babylonian mathematicians around 2000 BCE, making it one of the oldest mathematical results still in common use.
Quadratic equations model projectile motion: height = −½gt² + v₀t + h₀ where solving for t gives the time of flight.

Common mistakes to avoid

Sign errors in b — the formula uses −b, so a negative b becomes positive.
Forgetting to take the square root of the discriminant — compute √(b²−4ac), not b²−4ac.
Dividing by 2a not just 2 — the entire expression (−b ± √Δ) is divided by 2a.

Mathematical calculations are deterministic — results are exact for the inputs provided. Floating-point arithmetic in digital calculators may introduce small rounding errors for very large or very small coefficients. This calculator is for educational purposes.

Frequently Asked Questions

What is the quadratic formula?

The quadratic formula is x = (−b ± √(b² − 4ac)) / 2a. It gives the roots (solutions) of any equation in the form ax² + bx + c = 0.

What does the discriminant tell you?

The discriminant (b² − 4ac) tells you the nature of the roots. Positive = 2 real roots; zero = 1 repeated root; negative = 2 complex (imaginary) roots.

What is the vertex of a parabola?

The vertex is the highest or lowest point of the parabola, located at x = −b/2a, y = c − b²/4a.

Can a quadratic have no real solution?

Yes — when the discriminant is negative (b² − 4ac < 0), the roots are complex numbers involving the imaginary unit i.

How do I find the axis of symmetry?

The axis of symmetry is the vertical line x = −b / (2a), which passes through the vertex of the parabola.