Quadratic Equation Calculator
Solve ax² + bx + c = 0 with the quadratic formula, including the discriminant and complex roots.
How to use this calculator
- Rewrite your equation in the form ax² + bx + c = 0.
- Enter a, b, and c (including their signs).
- The calculator shows the discriminant and all roots.
The formula
x = (−b ± √(b² − 4ac)) ÷ 2a- a
- Coefficient of x² (cannot be zero)
- b
- Coefficient of x
- c
- Constant term
- b² − 4ac
- The discriminant — its sign decides how many real roots exist
How it works
The quadratic formula is derived by 'completing the square' on the general equation, and it works for every quadratic.
The discriminant tells the story before you solve: positive means two distinct real roots, zero means one repeated root, negative means two complex roots.
Worked example
Solve x² − 3x − 4 = 0.
- 1a = 1, b = −3, c = −4
- 2Discriminant = 9 + 16 = 25; √25 = 5
- 3x = (3 ± 5) / 2
x = 4 or x = −1. Check: (x − 4)(x + 1) = x² − 3x − 4. ✓
When this calculator is useful
- Algebra homework checking
- Projectile motion problems
- Finding break-even points
- Geometry optimization
Frequently asked questions
What if my equation isn't in standard form?
Move every term to one side first. For example, x² = 5x − 6 becomes x² − 5x + 6 = 0, so a = 1, b = −5, c = 6.
What does a negative discriminant mean?
The equation has no real solutions — graphically, the parabola never touches the x-axis. The calculator reports the complex roots.