Quadratic Formula Calculator
Solve ax² + bx + c = 0 for the discriminant, both roots, and the vertex
Enter a, b, and c for ax² + bx + c = 0. The calculator shows the discriminant, both roots, and the vertex. For a right triangle instead of a parabola, use the Pythagorean theorem calculator.
Calculator Inputs
Coefficient of x². Cannot be 0.
Coefficient of x
Constant term
Results
Enter values above to see your results
How to use this calculator
- Write the equation in standard form, ax² + bx + c = 0.
- Enter a, b, and c. Keep the signs. A missing x term means b is 0.
- Read the discriminant, the roots, and the vertex.
How the calculation works
The discriminant is b² − 4ac. Roots are (−b ± √discriminant) ÷ (2a). A positive discriminant gives two real roots. Zero gives one repeated real root. A negative discriminant gives two complex roots.
The vertex x-coordinate is −b ÷ (2a). The y-coordinate is the equation evaluated at that x.
What the discriminant tells you
For x² − 3x + 2 = 0, a is 1, b is −3, and c is 2. The discriminant is 9 − 8 = 1, so the roots are real and different: 2 and 1. The vertex is (1.5, −0.25).
- Discriminant greater than 0: two real roots.
- Discriminant equal to 0: one real root.
- Discriminant less than 0: two complex roots.
Worked example
x² − 3x + 2 = 0
(−3)² − 4(1)(2)= 1(3 ± √1) ÷ 2= 2 and 1x = 3 ÷ 2= (1.5, −0.25)Tips & Best Practices
- •Move every term to one side before you read a, b, and c.
- •If the equation is linear, a is 0 and the quadratic formula does not apply.
Frequently Asked Questions
What is the quadratic formula?
For ax² + bx + c = 0, each root is (−b ± √(b² − 4ac)) ÷ (2a), and a cannot be 0.
What if the discriminant is negative?
The roots are complex. The real part is −b ÷ (2a), and the imaginary part comes from the square root of the positive discriminant divided by 2a.
How do I find the vertex?
The x-coordinate is −b ÷ (2a). Substitute that x back into ax² + bx + c for the y-coordinate.