Quadratic Formula Calculator
Solve any quadratic equation ax² + bx + c = 0 with the quadratic formula, including complex roots, and see the discriminant, exact answers, vertex and every step
Last updated: November 12, 2026
How to Use the Quadratic Formula Calculator
How to use the quadratic formula calculator
- Write your equation in standard form. Move every term to one side so it reads ax² + bx + c = 0. For example, 2x² = 3x + 5 becomes 2x² − 3x − 5 = 0.
- Enter a, b and c. Include the signs. If a term is missing, enter 0, so x² − 9 = 0 has a = 1, b = 0 and c = −9.
- Read the solutions. You get the roots in decimal form, the exact form with fractions or square roots when the coefficients are whole numbers, and complex roots when the discriminant is negative.
- Check the extras. The calculator also shows the discriminant, vertex, axis of symmetry, vertex form, factored form when it is tidy, and the working step by step.
Decimals are rounded to 6 places. Use the exact form when your teacher asks for answers in simplest radical form.
Formula & Methodology
The quadratic formula
Any quadratic equation in standard form, ax² + bx + c = 0 with a ≠ 0, can be solved with:
x = (−b ± √(b² − 4ac)) ÷ 2a
The "±" means there are usually two answers: one using + and one using −.
The discriminant
The part under the square root, D = b² − 4ac, is called the discriminant. It tells you what kind of roots to expect before you finish the calculation:
- D > 0: two different real roots
- D = 0: one repeated real root (a double root)
- D < 0: no real roots, but two complex roots written as p ± qi
Worked examples
Two real roots. Solve x² − 3x + 2 = 0. Here a = 1, b = −3 and c = 2, so D = 9 − 8 = 1. Then x = (3 ± 1) ÷ 2, which gives x = 2 and x = 1.
Irrational roots. Solve x² − 3x + 1 = 0. D = 9 − 4 = 5, so x = (3 ± √5) ÷ 2, about 2.618 and 0.382.
Complex roots. Solve x² + 2x + 5 = 0. D = 4 − 20 = −16, and √−16 = 4i, so x = (−2 ± 4i) ÷ 2 = −1 ± 2i.
The vertex of the parabola
The graph of y = ax² + bx + c is a parabola. Its turning point, the vertex, is at:
x = −b ÷ 2a, and y = c − b² ÷ 4a
The vertical line through the vertex is the axis of symmetry. If a is positive the parabola opens upward and the vertex is the lowest point, and if a is negative it opens downward and the vertex is the highest point. Writing the equation as y = a(x − h)² + k puts it in vertex form, with the vertex at (h, k).
Sum and product of the roots
For any quadratic, the two roots add up to −b ÷ a and multiply to c ÷ a. This is a quick way to check your answers.
About Quadratic Formula Calculator
Our quadratic formula calculator is a free, easy-to-use tool designed to help yousolve any quadratic equation ax² + bx + c = 0 with the quadratic formula, including complex roots, and see the discriminant, exact answers, vertex and every step. If you're a professional or just need quick calculations, our calculator provides accurate results instantly.
Frequently Asked Questions
What is the quadratic formula?
It is x = (−b ± √(b² − 4ac)) ÷ 2a, which gives the solutions of any equation of the form ax² + bx + c = 0 where a is not zero. It works even when the equation cannot be factored.
How do I find a, b and c?
Rearrange the equation so one side is 0 and the terms are in the order x², x, then the constant. The number in front of x² is a, the number in front of x is b, and the constant is c. Keep the signs, so 3x² − 5x − 2 = 0 has a = 3, b = −5 and c = −2.
What does the discriminant tell you?
The discriminant b² − 4ac shows the number and type of solutions. A positive value means two real roots, zero means one repeated real root, and a negative value means two complex roots and no real ones.
What if the discriminant is negative?
Then the equation has no real solutions, and the parabola never crosses the x-axis. The roots are complex numbers of the form p ± qi, where i is the square root of −1. For x² + 2x + 5 = 0 they are −1 ± 2i.
Can the quadratic formula be used when a = 0?
No. If a is 0 the x² term disappears and the equation is linear, bx + c = 0, with the single solution x = −c ÷ b. The quadratic formula would divide by zero.
When should I factor instead of using the formula?
Factoring is quicker when the roots are whole numbers or simple fractions, such as x² − 5x + 6 = (x − 2)(x − 3). The quadratic formula always works, including for irrational and complex roots, so it is the safe choice when factoring is not obvious.
How do I find the vertex of a parabola?
The x-coordinate of the vertex is −b ÷ 2a. Substitute it back into the equation to get the y-coordinate, or use y = c − b² ÷ 4a. For y = x² − 4x + 1 the vertex is at (2, −3).
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