How the solver works
The tool first finds the discriminant D = b² − 4ac. When D is positive it computes the two real roots in a numerically stable way, which avoids losing accuracy when b is large. When D is negative it shows the complex roots as a real part plus or minus an imaginary part.
More than just the roots
You also get the axis of symmetry, the vertex and the sum and product of the roots (−b/a and c/a), which are useful for checking homework. Fractions in your answers? Use the fraction calculator. For very large or tiny numbers see the scientific notation converter.
Frequently asked questions
What is the quadratic formula?
For ax² + bx + c = 0 with a ≠ 0, x = (−b ± √(b² − 4ac)) ÷ 2a. The part under the root, b² − 4ac, is called the discriminant.
What does the discriminant tell me?
If it is positive there are two different real roots. If it is zero there is one repeated real root. If it is negative there are no real roots, and the two roots are complex numbers in the form p ± qi.
What if a is zero?
Then the equation is not quadratic. If b is not zero it becomes the straight-line equation bx + c = 0 with the single solution x = −c ÷ b, which this tool also solves.
What is the vertex?
The vertex is the turning point of the parabola y = ax² + bx + c. It sits at x = −b ÷ 2a. It is the minimum point if a is positive and the maximum if a is negative.
How can I check my answer?
Put each root back into the equation. If ax² + bx + c comes to zero (allowing for tiny rounding), the root is correct. For roots 2 and 3 of x² − 5x + 6, both 4 − 10 + 6 and 9 − 15 + 6 equal 0.