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Quadratic Equation Solver

Solve an equation of the form ax² + bx + c = 0 in seconds. Enter a, b and c and this free quadratic equation solver finds the real or complex roots, showing the discriminant so you understand exactly how it arrived at the answer — all privately in your browser.

Coefficient a

Coefficient b

Coefficient c

x = 2, x = 1

What is a quadratic equation solver?

A quadratic equation solver finds the values of x that make an equation of the form ax² + bx + c = 0 true. It uses the quadratic formula x = (−b ± √(b² − 4ac)) / 2a, where a, b and c are the coefficients of the quadratic, linear and constant terms. The calculator evaluates the discriminant — b² − 4ac — which tells you whether the roots are real, repeated or complex.

The formula's behaviour depends entirely on the discriminant. If it is positive, there are two distinct real roots; if it is zero, there is one repeated real root; and if it is negative, the roots are complex numbers with an imaginary part. For example, x² − 3x + 2 has a discriminant of 1, giving the roots 2 and 1.

This tool runs locally in your browser, showing the discriminant and the roots for any equation you type.

How to use the quadratic equation solver

Enter the coefficient a of the x² term.
Enter the coefficient b of the x term.
Enter the constant c.
Click Solve to see the discriminant and the roots.

Example

Enter a = 1, b = −3 and c = 2. The discriminant is (−3)² − 4(1)(2) = 1, which is positive, so there are two real roots: 2 and 1. Enter a = 1, b = 2, c = 5 and the discriminant becomes −16, so the roots are complex.

Who is this quadratic equation solver for?

A quadratic equation solver finds the roots of any equation in the form ax² + bx + c = 0 using the quadratic formula. It is a staple in classrooms and coursework.

Students: check the roots of your algebra and calculus homework in seconds, and see the steps behind the answer.
Teachers: set and verify quadratic problems, and show how the discriminant decides the number of roots.
College learners: solve physics and engineering problems where a parabola or projectile motion needs a solution.

Quadratic equation table

Using the quadratic formula x = (−b ± √(b² − 4ac)) ÷ 2a, these are the roots for a few common equations:

Equation	a	b	c	Discriminant	Roots
x² − 5x + 6 = 0	1	−5	6	1.00	2.00, 3.00
x² − 7x + 10 = 0	1	−7	10	9.00	2.00, 5.00
x² + 2x + 1 = 0	1	2	1	0.00	−1.00 (one root)
2x² − 3x + 1 = 0	2	−3	1	1.00	0.50, 1.00
3x² − 10x + 3 = 0	3	−10	3	64.00	0.33, 3.00

Common use cases

Solving quadratic problems in algebra, calculus and physics homework.
Finding the roots of a parabola to plot or analyse a function.
Working out where a projectile lands in a motion problem.
Checking work done by hand or in a spreadsheet.

Pro tips

Make sure a is not zero — if it is, the equation is linear, not quadratic.
The discriminant tells you what kind of roots to expect before you do any more work.
Rounding roots to four decimals is handled for you, so the answer stays readable.
Use the same values to check your own hand calculation.

FAQ

What is the quadratic formula?

x = (−b ± √(b² − 4ac)) / 2a, used to find the roots of any equation of the form ax² + bx + c = 0.

What do the three discriminant cases mean?

A positive discriminant gives two real roots, zero gives one repeated real root, and a negative discriminant gives two complex roots.

What if a is zero?

The equation is no longer quadratic — it becomes linear. The calculator tells you to enter a non-zero value for a.

What is the discriminant?

It is b² − 4ac, the part of the quadratic formula under the square root that determines the nature of the roots.

Can it solve complex roots?

Yes. When the discriminant is negative, the calculator returns complex roots with an imaginary part.

Is this calculator free?

Yes, it is completely free with no signup, and it runs entirely in your browser.

Does it show the working?

It shows the discriminant so you can see how the nature of the roots was determined.

What are the roots used for?

Roots are the x-intercepts of the parabola, where the function crosses or touches the x-axis.

How accurate are the results?

Roots are shown to four decimal places, which is plenty precise for most problems.