Quadratic Equation Calculator

What this calculator does

This calculator solves quadratic equations of the form ax² + bx + c = 0 using the quadratic formula. It finds the roots (solutions), determines whether they are real or complex, and provides the vertex, axis of symmetry, and discriminant of the corresponding parabola.

Quadratic equations appear throughout mathematics, physics, engineering, and economics. They describe projectile motion, optimize areas and revenues, model parabolic curves, and arise in countless textbook problems. This calculator handles all three cases—two distinct real roots, one repeated root, and complex (imaginary) roots—giving you complete solutions with step-by-step breakdowns.

Whether you are a student checking homework, a teacher preparing examples, or an engineer solving a design problem, this tool provides instant, accurate results.

How to use it

1. Enter the coefficients a, b, and c from your equation ax² + bx + c = 0.
2. The calculator computes the discriminant, roots, vertex, and axis of symmetry.
3. Review whether the roots are real and distinct, real and equal, or complex conjugates.

Formulas used

• Quadratic formula: x = (−b ± √(b² − 4ac)) / (2a)

• Discriminant (Δ): Δ = b² − 4ac

  • Δ > 0 → Two distinct real roots
  • Δ = 0 → One repeated real root (the parabola touches the x-axis)
  • Δ < 0 → Two complex conjugate roots (no real x-intercepts)

• Vertex of the parabola: x-coordinate = −b / (2a) y-coordinate = c − b² / (4a) (or substitute x back into ax² + bx + c)

• Axis of symmetry: x = −b / (2a)

Examples

Example 1 — Two real roots: 2x² − 7x + 3 = 0. a = 2, b = −7, c = 3. Δ = 49 − 24 = 25 (positive). √25 = 5. x = (7 ± 5) / 4 → x₁ = 3, x₂ = 0.5.

Example 2 — One repeated root: x² − 6x + 9 = 0. a = 1, b = −6, c = 9. Δ = 36 − 36 = 0. x = 6 / 2 = 3 (double root). The parabola's vertex is at (3, 0).

Example 3 — Complex roots: x² + 2x + 5 = 0. a = 1, b = 2, c = 5. Δ = 4 − 20 = −16 (negative). x = (−2 ± √(−16)) / 2 = −1 ± 2i.

Tips and common mistakes

• a cannot be zero. If a = 0, the equation is linear (bx + c = 0), not quadratic. The quadratic formula does not apply.
• Watch the sign of b. The formula uses −b, so a negative b becomes positive in the numerator. Sign errors are the most common mistake in manual calculation.
• Complex roots come in conjugate pairs. If one root is −1 + 2i, the other is always −1 − 2i. This is a property of polynomials with real coefficients.
• The discriminant tells you everything about the nature of the roots before you compute them. Check Δ first to know what to expect.
• Factoring is faster when possible. If the equation factors neatly (e.g. x² − 5x + 6 = (x−2)(x−3)), factoring is quicker than the formula. Use the calculator to verify or when factoring is not obvious.
• Vertex form is useful for graphing: y = a(x − h)² + k, where (h, k) is the vertex. The calculator provides these coordinates directly.