Quadratic Formula Calculator

How to use it

1. Enter a, b and c from ax² + bx + c = 0, or switch to Type the equation and write it as you see it, for example 3x^2 = 2x + 5 (terms can be on both sides; x² works too).
2. Coefficients can be whole numbers, decimals (0.25), fractions (3/4) or mixed numbers (1 1/2).
3. Read the exact roots, their decimal values, the discriminant, the vertex, the factored form when there is one, and the worked steps under the graph.

How the answer is worked out

The calculator first clears fractions and decimals by multiplying the whole equation by the smallest number that makes every coefficient a whole number, then divides out any common factor. That changes nothing about the roots but keeps every later number exact. It then computes the discriminant b² − 4ac with whole-number arithmetic of any size.

If the discriminant is a perfect square, both roots are fractions, and the equation factors: x² − 5x + 6 = 0 gives x = 2 or x = 3 and (x − 2)(x − 3). If not, the square root is simplified by pulling out square factors (√8 = 2√2, √72 = 6√2), and the whole fraction is reduced, so 2x² − 4x + 1 = 0 gives (2 ± √2)/2 rather than (4 ± 2√2)/4. A negative discriminant gives two complex roots with i: x² + 2x + 5 = 0 gives −1 ± 2i. When a is 0 the equation is linear, and the calculator says so and solves it.

Why the decimals do not lose digits

Typed straight into a computer, the quadratic formula can give a badly wrong small root. For x² + 100,000,000x + 1 = 0, the true roots are about −99,999,999.99999999 and −0.00000001. In ordinary double-precision arithmetic, (−b + √(b² − 4ac)) / 2a subtracts two nearly equal numbers and returns −7.45×10⁻⁹ for the small root, 25% off. This calculator works with 40-digit whole-number square roots, takes the larger root from the formula with the sign that adds rather than subtracts, and gets the smaller one from the product of the roots (c/a divided by the larger root), the standard fix described in Numerical Recipes and in Goldberg’s paper on floating-point arithmetic. It shows −0.00000001 for that root.

What we tested

On October 15, 2026 we ran the calculator’s own code on 2,912 equations and checked every answer with SymPy 1.14.0, a computer algebra system: random whole numbers, fractions, mixed numbers, decimals and 12-digit coefficients, plus equations built from chosen roots. Each exact root was substituted back into the equation symbolically, each decimal was compared with SymPy at 60 digits, and each vertex and factored form was expanded back.

Failures: 0. The equation box read 15 of 15 test equations correctly (terms on both sides, x², fractions, decimals, capital X) and refused all 8 malformed ones (x^3, brackets, two = signs, another letter) with a message. In Chromium the page also gave the same answers on a 390-pixel phone screen and kept the last equation after a reload.

Tips

1. Check the discriminant first: positive means two real roots, zero means one, negative means none on the real number line (the graph will not touch the x-axis).
2. The vertex is the turning point; the axis of symmetry is the vertical line through it, halfway between the roots.
3. If your teacher wants the answer in a different but equal form, (2 ± √2)/2 is the same as 1 ± √2/2.
4. Brackets are not expanded for you: (x + 1)² = 4 becomes x² + 2x + 1 = 4 before you type it.

Frequently Asked Questions

Built by Derek Giordano · Part of Ultimate Design Tools

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