Quadratic Equation
Definition
A second-degree polynomial equation of the form ax² + bx + c = 0, solvable using the quadratic formula: x = (-b ± √(b²-4ac)) / 2a.
Why is Quadratic Equation Important?
Quadratic Equation is a foundational mathematical concept used across science, engineering, finance, and everyday problem-solving. From analyzing data sets to optimizing business decisions, this concept provides the analytical framework needed to interpret quantitative information accurately.
Our math calculators make complex computations simple and accessible, providing step-by-step results that help students, professionals, and curious minds explore mathematical relationships with confidence.
What is a Quadratic Equation?
A quadratic equation is a second-degree polynomial equation of the form ax² + bx + c = 0, where a ≠ 0. It produces a parabola when graphed. Every quadratic equation has exactly two solutions (which may be real or complex, and may be equal).
The Quadratic Formula
x = (−b ± √(b² − 4ac)) / 2a
The Discriminant (Δ = b² − 4ac)
| Discriminant Value | Meaning | Graph |
|---|---|---|
| Δ > 0 | Two distinct real solutions | Parabola crosses x-axis twice |
| Δ = 0 | One repeated real solution | Parabola touches x-axis once (vertex) |
| Δ < 0 | Two complex conjugate solutions | Parabola doesn't cross x-axis |
Solving Methods
| Method | Best For | Example: x² − 5x + 6 = 0 |
|---|---|---|
| Factoring | Simple integer solutions | (x−2)(x−3) = 0 → x = 2 or 3 |
| Quadratic Formula | Any quadratic equation | x = (5 ± √(25−24))/2 = (5±1)/2 → 2 or 3 |
| Completing the Square | Deriving vertex form | (x − 5/2)² = 1/4 → x = 5/2 ± 1/2 |
Real-World Applications
- Projectile motion: h(t) = −16t² + v₀t + h₀ (height as function of time)
- Business: Profit = −ax² + bx − c (profit maximization at vertex)
- Engineering: Cable sag, arch design, trajectory calculations
- Physics: Kinematic equations for uniformly accelerated motion