Quadratic formula: solving second-degree equations
A quadratic equation has the form ax² + bx + c = 0, where a ≠ 0. The quadratic formula solves any equation of this type and determines its real or complex roots.
The quadratic formula
x = (−b ± √(b² − 4ac)) / 2a. The two ± signs produce the two possible solutions (x₁ and x₂). The expression under the root, b² − 4ac, is called the discriminant and determines the nature of the solutions.
The discriminant and solution types
- Δ > 0: two distinct real solutions. The parabola crosses the X axis at two points.
- Δ = 0: one double real solution. The parabola is tangent to the X axis.
- Δ < 0: no real solutions. Roots are complex conjugate numbers.
Real-life applications
Quadratic equations model parabolic trajectories (projectiles, arches), areas and dimensions of geometric figures, compound interest calculations, and optimization problems in physics, engineering, and economics.