Quadratic Equation Calculator

Solve ax² + bx + c = 0 with the quadratic formula, and check the answer instantly: the roots must add to −b/a and multiply to c/a.

Solutions

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Solutions—

Discriminant—

Type of roots—

Sum of the roots—

Product of the roots—

Factorised form—

How it was solved

The formula

x = (−b ± √(b² − 4ac)) ÷ 2a

Why it works

The quadratic formula comes from completing the square on the general equation, which is why it works for every quadratic without exception. The part under the root, b² − 4ac, is the discriminant, and it alone decides whether there are two real solutions, one, or none.

How to solve it by hand

Work out the discriminant: b² − 4ac
If it is negative there are no real solutions; stop there
Take its square root
Apply (−b ± that root) ÷ 2a for the two solutions

What is worth knowing

Vieta's formulas give you a free check that takes two seconds: the roots always add up to −b/a and multiply to c/a. If your answers fail either test, you have made an arithmetic slip. They are also how factorising by inspection works — when you look for two numbers that add to b and multiply to c, you are using Vieta's without knowing it. The other thing worth internalising is that a negative discriminant is not a failure: the solutions exist, they are just complex, and they always come in conjugate pairs.

Frequently asked questions

What does the discriminant tell you?

Positive means two distinct real roots, zero means one repeated root, and negative means no real roots — the solutions are complex.

How can I check my answers?

Add them: they should give −b/a. Multiply them: they should give c/a. Both are Vieta's formulas and both are quick.

When can I factorise instead?

When the roots are simple rationals. Looking for two numbers that add to b and multiply to c is exactly that, and the formula always works when it fails.

What if a is zero?

Then it is not a quadratic but a linear equation, and the formula would divide by zero. The calculator says so and solves it as linear.