What is the quadratic formula?

It is x = (−b ± √(b² − 4ac)) ÷ 2a, and it gives both roots of any equation written as ax² + bx + c = 0. The ± sign is why a quadratic usually has two solutions. This calculator fills in your a, b and c and shows each step.

What does the discriminant tell me?

The discriminant is b² − 4ac. If it is positive there are two different real roots, if it is zero there is one repeated real root, and if it is negative the two roots are a complex conjugate pair. The calculator labels which case you are in.

Can it give complex roots?

Yes. When the discriminant is negative the equation has no real roots, but it does have two complex roots like −1 + 2i and −1 − 2i. The calculator works these out instead of just reporting no solution.

Quadratic Formula Calculator

Enter the coefficients a, b and c to solve ax² + bx + c = 0. You get both roots, the discriminant, and the complex roots when the parabola never crosses zero. The working is shown line by line. A free quadratic formula calculator that runs in your browser.

Solve ax² + bx + c = 0 with the quadratic formula by entering a, b and c. You get both roots, the discriminant, and the complex roots when there are none on the real line, with the working shown line by line. It answers how to solve a quadratic whether the parabola crosses zero twice, once or not at all.

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abc

For the equation a·x² + b·x + c = 0.

Roots of 1x² + -5x + 6 = 0

x = 3, 2

x = (−b ± √(b² − 4ac)) ÷ 2a = (−(-5) ± √(1)) ÷ 2

Discriminant (b² − 4ac)

Nature

Two real

Axis of symmetry

2.5

Two distinct real roots.

How to use it

1
Enter a, b and c

Type the three coefficients from your equation in the form ax² + bx + c = 0. The value of a cannot be zero, since that makes the equation linear.
2
Read the discriminant

The calculator shows b² − 4ac first, because its sign tells you whether the roots are two real, one repeated, or a complex pair.
3
Read both roots

See x₁ and x₂ worked out from the formula, in exact form where it is clean and as decimals otherwise.

When it comes in handy

Algebra homework

Solve a quadratic and check your roots against the discriminant so you understand why there are two answers, one, or none on the real line.

Physics and projectiles

Find when a height or distance reaches zero, which is a quadratic in time for anything thrown or dropped.

Checking by hand

Confirm a result you got with the formula yourself, including the complex case where sign mistakes are common.

Instant, exact & 100% in your browser

The maths runs right here in your browser, with fractions and whole numbers kept exact rather than rounded along the way. Nothing you type is sent to a server, there is no sign-up and no limit, and once the page has loaded it keeps working even with no connection.

Frequently asked questions

What is the quadratic formula?: It is x = (−b ± √(b² − 4ac)) ÷ 2a, and it gives both roots of any equation written as ax² + bx + c = 0. The ± sign is why a quadratic usually has two solutions. This calculator fills in your a, b and c and shows each step.
What does the discriminant tell me?: The discriminant is b² − 4ac. If it is positive there are two different real roots, if it is zero there is one repeated real root, and if it is negative the two roots are a complex conjugate pair. The calculator labels which case you are in.
Can it give complex roots?: Yes. When the discriminant is negative the equation has no real roots, but it does have two complex roots like −1 + 2i and −1 − 2i. The calculator works these out instead of just reporting no solution.

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How to use it

Enter a, b and c

Read the discriminant

Read both roots