Quadratic Formula Calculator
Solve any quadratic equation of the form ax² + bx + c = 0. Enter the coefficients a, b, and c to find both roots (real or complex), the discriminant, and a step-by-step solution.
About the Quadratic Formula
The quadratic formula solves any quadratic equation ax² + bx + c = 0:
The discriminant (b² − 4ac) tells you about the roots:
- > 0: Two distinct real roots
- = 0: One repeated real root (double root)
- < 0: Two complex (imaginary) roots
Frequently Asked Questions
If a = 0, the equation is linear (bx + c = 0), not quadratic. The solution is simply x = −c/b.
When the discriminant is negative, the roots involve the square root of a negative number. These are written as complex numbers with a real part and an imaginary part (e.g., 2 + 3i).
The formula
It solves any quadratic equation. The ± sign is why there are normally two answers — one using the plus, one using the minus.
Worked example
Solve x² − 5x + 6 = 0, so a = 1, b = −5, c = 6.
- b² − 4ac = 25 − 24 = 1
- x = (5 ± 1) ÷ 2
- x = 3 or x = 2
Check by substituting back: 9 − 15 + 6 = 0 and 4 − 10 + 6 = 0. Both work.
The discriminant tells you what to expect
The part under the square root, b² − 4ac, is called the discriminant, and its sign determines the nature of the solutions before you finish the calculation:
| Discriminant | Solutions |
|---|---|
| Positive | Two distinct real solutions |
| Zero | One repeated real solution |
| Negative | No real solutions (two complex) |
Graphically this is where the parabola meets the x-axis: twice, once at the vertex, or not at all.
Common mistakes
- Losing the sign of b. In x² − 5x + 6, b is −5, so −b is +5. Dropping the negative is the most frequent error.
- Dividing only part of the numerator. The 2a divides the entire expression, not just the square root term.
- Not rearranging first. The equation must equal zero. For x² = 5x − 6, move everything across to get x² − 5x + 6 = 0 before identifying a, b and c.
When not to use it
If the equation factors easily, factoring is quicker. x² − 5x + 6 = (x − 3)(x − 2) gives the same answer in one line. The formula's advantage is that it always works, including for equations that don't factor into neat whole numbers.
Quadratics model projectile paths, area problems where one dimension depends on another, and break-even points where revenue and cost curves cross.