How to use this calculator
- Write your equation in the form ax² + bx + c = 0.
- Enter a, b and c (a cannot be 0). Use negative numbers for subtracted terms.
- Read the roots and other properties.
Formula
The discriminant Δ = b² − 4ac tells you the type of roots: positive gives two real roots, zero gives one repeated root and negative gives two complex conjugate roots. The calculator uses a numerically stable form of the formula to avoid rounding loss.
x = (−b ± √(b² − 4ac)) / 2a
Vertex: (−b/2a, c − b²/4a)
Example
- x² − 3x − 4 = 0: a = 1, b = −3, c = −4.
- Δ = 9 + 16 = 25; √Δ = 5.
- x = (3 ± 5) / 2, so x = 4 or x = −1.
Frequently asked questions
What if a = 0?
Then the equation is linear (bx + c = 0), not quadratic, and its solution is x = −c/b.
What are complex roots?
When Δ < 0, the parabola doesn't cross the x-axis. The roots involve i = √−1 and come as a conjugate pair a ± bi.
How else can I solve a quadratic?
By factoring (when the roots are rational), completing the square, or graphing. The formula always works.