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Quadratic Formula Calculator

Enter the coefficients a, b and c of ax² + bx + c = 0 and press Solve. The calculator returns both roots, the discriminant, vertex and axis of symmetry.

Quick Presets:

Quadratic Equation: ax² + bx + c = 0

Roots & Parabola Analysis

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Formula: x = (-b ± √(b² - 4ac)) / 2a
Discriminant: D = b² - 4ac
Roots: Two real, one repeated, or two complex
Extras: Vertex and axis of symmetry

The quadratic formula explained

A quadratic equation has the form ax² + bx + c = 0. Its solutions are given by x = (-b ± √(b² - 4ac)) / 2a. The expression under the square root, b² - 4ac, is called the discriminant.

The discriminant decides the nature of the roots: if it is positive there are two distinct real roots; if zero, exactly one repeated real root; if negative, two complex roots. The calculator handles all three cases automatically.

The vertex and axis of symmetry

The graph of a quadratic is a parabola. Its vertex, the highest or lowest point, sits at x = -b / (2a), and the vertical line through that point is the axis of symmetry.

The value of a tells you the direction: a positive a opens upward (vertex is a minimum), a negative a opens downward (vertex is a maximum). These details are included in the results.

How to use this calculator

  1. Rewrite your equation in the form ax² + bx + c = 0.
  2. Enter the coefficients a, b and c.
  3. Press Solve to see the roots.
  4. Read the discriminant, vertex and axis of symmetry for the graph.

Pro tips

  • Move every term to one side before reading off the coefficients.
  • If a is 0 the equation is linear, not quadratic — this calculator requires a ≠ 0.
  • A perfect square discriminant gives rational roots that you can verify by factoring.

Advantages & considerations

✅ Advantages

  • Handles real, repeated and complex roots
  • Shows discriminant, vertex and axis
  • Works with decimal coefficients

⚠️ Considerations

  • Complex roots are displayed as a ± bi pairs
  • Requires a non-zero leading coefficient

Frequently asked questions

What is the quadratic formula?▶
For ax² + bx + c = 0, the roots are x = (-b ± √(b² - 4ac)) / 2a. It works for every quadratic, whether the roots are real or complex.
What does the discriminant tell me?▶
If b² - 4ac is positive, two real roots; if zero, one repeated root; if negative, two complex roots. It tells you how the parabola crosses the x-axis.
How do I find the vertex of a parabola?▶
The x-coordinate is -b / (2a). Plug it back into the equation to get the y-coordinate. The calculator reports both.
Can I solve a quadratic with decimals?▶
Yes. Enter decimal coefficients such as a = 0.5, b = -2.25 and c = 3. The calculator solves exactly using floating-point math.
What are complex roots?▶
When the discriminant is negative, the square root is imaginary. The solutions then come in pairs a ± bi, such as -2 + i and -2 - i.