Math & School

Quadratic equation solver

Solve a quadratic equation – pq-formula or quadratic formula, with a worked example.

Updated on May 25, 2026 Calculator, calculation path and examples on one page

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Formula

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Examples

x² − 6x + 8 = 0 → x = 4 and x = 2.

x₁: 4.00

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x² − 4x + 4 = 0 → x = 2.

Solution (double): x = 2.00

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2x² − 8x + 6 = 0 → x = 3 and x = 1.

x₁: 3.00

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FAQ

What is the pq formula?

It solves equations of the form x² + px + q = 0 with x = −p/2 ± √((p/2)² − q).

What is the midnight formula?

The abc or midnight formula solves ax² + bx + c = 0 with x = (−b ± √(b² − 4ac)) / (2a).

What does the discriminant say?

If D is greater than 0, there are two solutions; if D is 0, a double solution; if D is less than 0, not a real solution.

When do I use pq and when abc?

The pq formula fits if there is a 1 in front of the x². Otherwise the midnight formula with a, b and c is more convenient.

Can I put an equation with a in front of x² into pq form?

Yes, by dividing the whole equation by a. Or you can use the midnight formula directly.

Are negative or decimal numbers also allowed?

Yes. You can enter any real numbers for p, q, a, b and c.

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