What Is the Quadratic Formula?
The quadratic formula solves any equation of the form ax² + bx + c = 0 for x, where a, b, and c are known coefficients and a ≠ 0. It works for every quadratic equation, including ones that don't factor neatly.
How to Use the Quadratic Formula Calculator
- Enter the coefficients a, b, and c from your equation.
- The calculator solves instantly and shows the discriminant.
The Quadratic Formula
The Discriminant
The discriminant (b² − 4ac) tells you what kind of solutions to expect before you even finish solving:
| Discriminant | Result |
|---|---|
| Positive (> 0) | Two distinct real roots |
| Zero (= 0) | One repeated real root |
| Negative (< 0) | Two complex (imaginary) roots |
Worked Example
a = 1, b = −3, c = 2
Discriminant: (−3)² − 4(1)(2) = 9 − 8 = 1
x = (3 ± √1) / 2 = (3 ± 1) / 2
Solutions: x = 2 and x = 1
Understanding Your Results
The two ± solutions correspond to the two points where the parabola y = ax² + bx + c crosses the x-axis (its roots). When the discriminant is zero, the parabola just touches the x-axis at one point. When it's negative, the parabola never crosses the x-axis, and the solutions are complex numbers.
Common Mistakes to Avoid
- Forgetting the ± sign, which produces only one of the two solutions.
- Mixing up the sign of b when substituting into the formula (it's −b, not b).
- Dividing only part of the numerator by 2a instead of the entire expression.
Frequently Asked Questions
It solves any quadratic equation of the form ax² + bx + c = 0 for the value(s) of x, even when the equation doesn't factor into simple whole numbers.
It tells you the nature of the solutions before solving: positive means two real roots, zero means one repeated real root, and negative means two complex roots.
Yes — when the discriminant is negative, the square root of a negative number produces complex (imaginary) solutions, shown as a ± bi.
If a = 0, the equation isn't quadratic anymore — it's linear (bx + c = 0), and the quadratic formula doesn't apply since it would require dividing by zero.
It comes from completing the square on the general form ax² + bx + c = 0, a classic algebraic technique that isolates x.