Two real roots
With a = 1, b = −5, and c = 6, the discriminant is 1 and the roots are 2 and 3.
Calculations run locally in your browser. This tool does not upload the numbers you enter.
How to use the Quadratic Equation Solver
- 01
Move every term to the left so the equation has the form ax² + bx + c = 0.
- 02
Enter a, b, and c, including zero for a missing term. Use a minus sign for negative coefficients.
- 03
Calculate and read the root classification. Copy or download the equation and its approximate roots.
Examples
Complex conjugates
With a = 1, b = 2, and c = 5, the discriminant is −16. The roots are −1 + 2i and −1 − 2i.
Reading the discriminant
For nonzero a, the discriminant is b² − 4ac. A positive value gives two distinct real roots; zero gives a repeated real root; and a negative value gives a complex conjugate pair. In the complex result, i is the number satisfying i² = −1.
A more stable real-root calculation
For two distinct real roots, the solver computes the root whose numerator avoids subtraction of nearly equal values, then obtains the other from the product-of-roots identity c/a. Coefficients are scaled first. This reduces a common cancellation problem, but arithmetic and displayed values are still approximate.
When the equation is not quadratic
If a is zero and b is nonzero, the solver returns x = −c/b. If a and b are both zero, c = 0 makes every real number a solution; otherwise there is no solution. These cases are not forced into the quadratic formula.
What to know
- Numerical roots use floating-point arithmetic and 12-significant-digit display; extreme coefficient ratios can exceed the supported range.
Common questions
Can I enter fractions such as 1/3?
Enter a decimal approximation such as 0.333333333333, or multiply the entire equation by a denominator first. Expression parsing is not provided.
Why can near-repeated roots be sensitive?
When b² and 4ac nearly match, a tiny coefficient change or rounding error can change the sign of the discriminant. The displayed classification follows the supplied floating-point values.
Does it show radicals?
No. The tool returns numerical roots, rounded for display to 12 significant digits, rather than symbolic radical simplifications.
What does a repeated root mean?
Both quadratic-formula branches have the same value. For example, x² − 4x + 4 = 0 has the repeated root x = 2.