Enter the coefficients a, b, and c to solve a quadratic equation for x.

Solving quadratic equations

A quadratic equation has the form a times x squared plus b times x plus c equals zero, and the quadratic formula solves any of them. It is one of the most useful results in algebra, appearing in physics, finance, and geometry whenever a relationship curves rather than runs straight.

The part under the square root, called the discriminant, decides the outcome before you finish. A positive discriminant yields two real solutions, zero yields a single repeated solution, and a negative discriminant yields a pair of complex solutions, which the calculator reports in the form with i.

The formula

Worked example

For x^2 - 3x + 2 = 0, the roots are x = 2 and x = 1.

How to read your result

The discriminant b squared minus 4ac tells you the nature of the roots: positive gives two real roots, zero gives one repeated root, and negative gives a pair of complex roots.

Frequently asked questions

What if a is zero?

Then the equation is linear, not quadratic, and the formula does not apply.

What does the discriminant tell me?

Whether the roots are two real, one repeated, or two complex.

Can roots be complex?

Yes, when the discriminant is negative the calculator reports complex roots.

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