The quadratic formula
Any equation that can be written as ax² + bx + c = 0 (with a ≠ 0) is solved by:
x = (−b ± √(b² − 4ac)) ÷ 2a
The ± means there are usually two answers. For x² − 3x + 2 = 0, with a = 1, b = −3, c = 2, the discriminant is 9 − 8 = 1, so x = (3 ± 1) ÷ 2 — giving x = 2 and x = 1.
What the discriminant tells you
| Discriminant (b² − 4ac) | Roots |
|---|---|
| Positive | Two different real roots |
| Zero | One repeated real root |
| Negative | Two complex (imaginary) roots |
Complex roots
When the discriminant is negative, the roots are complex conjugates of the form p ± qi. They still satisfy the equation — they just don't appear where the parabola crosses the x-axis, because it doesn't.
Frequently asked questions
What is the quadratic formula?
It solves any equation of the form ax² + bx + c = 0. The roots are x = (−b ± √(b² − 4ac)) ÷ 2a. The ± sign gives the two solutions. For x² − 3x + 2 = 0, the roots are x = 2 and x = 1.
What is the discriminant?
The discriminant is the part under the square root, b² − 4ac. Its sign tells you the type of roots: positive means two distinct real roots, zero means one repeated real root, and negative means two complex (imaginary) roots. This calculator shows the discriminant and the root type.
What if a equals zero?
Then the equation is not quadratic — it is linear (bx + c = 0), with a single solution x = −c ÷ b. The quadratic formula requires a to be non-zero, so the calculator flags this case.
What are complex roots?
When the discriminant is negative, the square root of a negative number is imaginary, so the roots take the form p ± qi, where i is the square root of −1. The two roots are complex conjugates — same real part, opposite imaginary parts.
Note: The coefficient a must be non-zero for a quadratic. If a = 0, the equation is linear and the calculator solves it as such.