Math & Statistics
Quadratic Formula
Solve any quadratic equation ax² + bx + c = 0.
The quadratic formula solves any second-degree polynomial equation. It appears everywhere from projectile motion in physics to profit maximization in economics.
Solutions
x₁ = 3.000000
- x1
- 3.000000
- x2
- 2.000000
- discriminant
- 1.0000
- vertex
- (2.5000, -0.2500)
Formula and method
x = (-b ± √(b² - 4ac)) / 2a
Direct application of the quadratic formula. Discriminant (b² - 4ac) determines the nature of roots: positive = 2 real roots, zero = 1 double root, negative = complex roots.
Worked examples
Classic factoring problem
- a (coefficient of x²):
- 1
- b (coefficient of x):
- -5
- c (constant):
- 6
Solutions: x₁ = 3.000000
No real solution
- a (coefficient of x²):
- 1
- b (coefficient of x):
- 0
- c (constant):
- 4
Discriminant < 0: No Real Solutions
Assumptions and limits
- Real coefficients only
- Standard form ax² + bx + c = 0
This tool provides an educational estimate. Rules, rates and personal circumstances can change the answer. Review the assumptions before relying on a result. Report a correction.
Common mistakes to avoid
- Forgetting to apply the ± correctly — the formula produces two roots.
- Using integer arithmetic — use decimals to avoid rounding errors in intermediate steps.
- Misidentifying coefficients from a non-standard form equation.
Frequently asked questions
What does the discriminant tell us?
b² - 4ac: if positive, two distinct real roots; if zero, one repeated real root; if negative, two complex (imaginary) roots. The sign of the discriminant describes the parabola's relationship with the x-axis.
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