Solve ax² + bx + c = 0
About this tool
Enter the coefficients a, b and c of a quadratic equation ax² + bx + c = 0 and this calculator finds the roots with the quadratic formula. It handles two real roots, one repeated root and complex roots, and also shows the discriminant and the vertex of the parabola. It runs in your browser and nothing is stored.
How to use it
Rewrite your equation so that one side is zero, then type the number in front of x² as a, the number in front of x as b, and the constant as c. For x² - 3x + 2 = 0, enter a = 1, b = -3 and c = 2. The roots appear immediately, here x = 2 and x = 1.
The quadratic formula
The solutions are x = (-b ± √(b² - 4ac)) / (2a). The part under the root, b² - 4ac, is called the discriminant, and it tells you what kind of roots to expect before you solve anything.
What the discriminant tells you
- Positive: two different real roots, where the parabola crosses the x-axis twice.
- Zero: one repeated real root, where the parabola just touches the x-axis.
- Negative: two complex roots, and the parabola never crosses the x-axis.
Worked example
Take 2x² + 5x - 3 = 0. Here a = 2, b = 5, c = -3, so the discriminant is 25 + 24 = 49 and its square root is 7. The roots are (-5 + 7) / 4 = 0.5 and (-5 - 7) / 4 = -3. The vertex is at x = -b / 2a = -1.25, where y = -6.125.
Other ways to solve a quadratic
Factoring is fastest when the numbers are small and the equation factors cleanly. Completing the square gives the vertex form and leads to the formula itself. The quadratic formula always works, which is why it is the safe default.
Things to watch out for
If a is zero the equation is linear, not quadratic, so there is only one solution, x = -c / b. Check the signs when you copy the coefficients, because a missing minus sign changes the answer. Roots shown to six decimals are rounded, so very close roots may look identical.
Frequently asked questions
For ax^2 + bx + c = 0, the solutions are x = (-b plus or minus the square root of (b^2 - 4ac)) divided by 2a.
The discriminant is b^2 - 4ac. If it is positive there are two real roots, if zero there is one repeated root, and if negative the roots are complex.
Then the equation is not quadratic. It becomes bx + c = 0 and has one solution, x = -c / b.