Quadratic Formula Calculator

Solve ax² + bx + c = 0 and see every step.

WhatMathsQUAD-01

Example

For x² - 3x + 2 = 0, a = 1, b = -3 and c = 2. The discriminant is (-3)² - 4(1)(2) = 1, so x = (3 ± 1) / 2, which gives x = 2 and x = 1.

What is the quadratic formula?

The quadratic formula solves any equation written as ax² + bx + c = 0, where a is not zero. The solutions are x = (-b ± √(b² - 4ac)) / 2a. Every quadratic equation has two solutions. They can be two different real numbers, one repeated real number, or a pair of complex numbers.

What the discriminant tells you

The expression under the square root, D = b² - 4ac, is called the discriminant. It tells you what kind of solutions to expect before you finish the calculation.

  • If D is greater than 0, there are two different real solutions and the parabola crosses the x-axis twice.
  • If D is exactly 0, there is one repeated real solution and the parabola touches the x-axis at its vertex.
  • If D is less than 0, there are no real solutions. The two solutions are complex numbers of the form p ± qi, and the parabola never touches the x-axis.

How to use this calculator

Type the three coefficients a, b and c. First write the equation in standard form, with every term on one side and 0 on the other. For example, x² = 5x - 6 becomes x² - 5x + 6 = 0, so a = 1, b = -5 and c = 6. The results update as you type and the steps panel shows the numbers substituted into the formula. The calculator also reports the vertex, the axis of symmetry and the y-intercept.

Worked example with a negative discriminant

Solve x² + 2x + 5 = 0. Here a = 1, b = 2 and c = 5, so D = 2² - 4(1)(5) = -16. The square root of -16 is 4i, so x = (-2 ± 4i) / 2 = -1 ± 2i. There are no real solutions.

Common mistakes

  • Forgetting to move every term to one side before reading off a, b and c.
  • Dropping the sign of b. When b is negative, -b is positive.
  • Dividing only part of the numerator by 2a. The whole of -b ± √D is divided by 2a.
  • Using the formula when a = 0. Then the equation is linear, and this calculator solves it as bx + c = 0.

Other ways to solve a quadratic

Factoring is quick when the numbers are small, completing the square is useful when you need vertex form, and graphing shows the solutions as x-intercepts. The quadratic formula is the one method that always works. To solve two equations at once, try the system of equations calculator.