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Quadratic Equation Calculator

Solve quadratic equations of the form ax² + bx + c = 0. Find real and complex roots using the quadratic formula.

Enter coefficients for ax² + bx + c = 0

What is Quadratic Equation Calculator?

A quadratic equation calculator solves equations of the form ax² + bx + c = 0 using the quadratic formula. It finds both real and complex roots, calculates the discriminant, and provides additional properties like the vertex, axis of symmetry, and sum/product of roots.

How to use

  1. 1 Enter the coefficients a, b, and c for your equation ax² + bx + c = 0.
  2. 2 The discriminant and roots (x₁ and x₂) appear instantly.
  3. 3 The nature of roots (two real, one repeated, or complex) is shown below the discriminant.
  4. 4 Scroll down to see equation properties including vertex coordinates and axis of symmetry.

Formula

Quadratic formula: x = (-b ± √(b² - 4ac)) / (2a). Discriminant: Δ = b² - 4ac. If Δ > 0: two real roots. If Δ = 0: one repeated root. If Δ < 0: two complex conjugate roots.

Example calculation

For x² - 5x + 6 = 0 (a=1, b=-5, c=6): Δ = 25 - 24 = 1. Roots: x = (5 ± 1) / 2, giving x₁ = 3 and x₂ = 2. These are the values where the parabola crosses the x-axis.

Frequently asked questions

What is the discriminant and why does it matter?

The discriminant (b² - 4ac) determines the nature of the roots. Positive means two distinct real roots; zero means one repeated real root; negative means two complex (imaginary) roots.

What if coefficient a is zero?

If a = 0, the equation becomes linear (bx + c = 0), not quadratic. The calculator will show an error in this case since the quadratic formula requires a ≠ 0.

What are complex roots?

Complex roots occur when the discriminant is negative. They take the form p ± qi where i = √(-1). They appear in conjugate pairs and mean the parabola doesn't cross the x-axis.

What is the vertex of a parabola?

The vertex is the highest or lowest point of the parabola. For ax² + bx + c, the vertex x-coordinate is -b/(2a) and the y-coordinate is c - b²/(4a).

Can I solve equations where the roots are fractions or irrationals?

Yes. The calculator handles all real and complex solutions, displaying results as decimals. For example, x² - 2 = 0 gives roots ±√2 ≈ ±1.4142136.