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Quadratic Formula Calculator

Solve ax² + bx + c = 0 with fractions allowed; roots, discriminant, axis, vertex, and derivation steps.

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How it works

Solves ax² + bx + c = 0 with x = (−b ± √(b² − 4ac)) / (2a). Shows completing-the-square steps, discriminant cases, and parabola features (axis, vertex, intercepts).

x = (−b ± √(b² − 4ac)) / (2a)

When to use

Use this tool when you need to: fractions OK (3/4). Example: a=1, b=−5, c=6 → roots 3 and 2. a cannot be zero.

How to use this tool

  1. Enter your values in the fields above.
  2. Review the formula and any mode options for your problem.
  3. Read the result and the step-by-step solution.

Fractions OK (3/4). Example: a=1, b=−5, c=6 → roots 3 and 2. a cannot be zero.

Frequently asked questions

What is the Quadratic Formula Calculator?
Solves ax² + bx + c = 0 with x = (−b ± √(b² − 4ac)) / (2a). Shows completing-the-square steps, discriminant cases, and parabola features (axis, vertex, intercepts).
How do I use the Quadratic Formula Calculator?
Enter the required values, then calculate. The Quadratic Formula Calculator shows results, formulas, and step-by-step work when available. No account is required.
What formula does the Quadratic Formula Calculator use?
x = (−b ± √(b² − 4ac)) / (2a)

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Quadratic formula

Solve ax² + bx + c = 0 for roots, discriminant, axis of symmetry, vertex, and Vieta checks. Fractions like 3/4 are allowed; a cannot be 0.

x2+x+=0

x² − 5x + 6 = 0

Use decimals or fractions (3/4, −1/2) in a, b, or c.

Try an example

How it works

What is a quadratic equation?

In algebra, a quadratic equation is any second-degree polynomial equation of the form: ax² + bx + c = 0 where x is unknown, a is the quadratic coefficient, b the linear coefficient, and c the constant. The value a cannot be 0, or the equation would be linear rather than quadratic. You can solve by factoring, the quadratic formula, completing the square, or graphing. This tool uses the quadratic formula and shows a completing-the-square derivation for every solve.

Derivation of the quadratic formula

Start from ax² + bx + c = 0 and divide by a: x² + (b/a)x + c/a = 0 Move the constant, then complete the square by adding (b/(2a))² to both sides. After taking square roots and solving for x you obtain: x = (−b ± √(b² − 4ac)) / (2a) The ± comes from the two square-root branches — both the positive and negative roots can be solutions.

Roots, the discriminant, and the axis

The x values from the formula are roots of the equation — the x-intercepts where the parabola y = ax² + bx + c crosses (or touches) the x-axis. Δ = b² − 4ac: Δ > 0 — two distinct real roots (two intercepts) Δ = 0 — one repeated real root (vertex on the axis) Δ < 0 — complex conjugate roots (no real intercepts) The axis of symmetry is x = −b/(2a); it also equals the average of the two roots when they are real. Example: x² − 5x + 6 = 0 → Δ = 1 → roots 3 and 2; axis x = 2.5.

Graph and applications

The vertex of y = ax² + bx + c is at (−b/(2a), f(−b/(2a))). The sign of a decides whether the parabola opens upward or downward. The y-intercept is c. By Vieta’s formulas, the sum of roots is −b/a and the product is c/a (counting complex roots the same way). Real-world uses include area problems, projectile trajectories, speed models, and other parabola-shaped relationships.

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