What Are Quadratic Equations?
A quadratic equation is a polynomial equation of degree two, typically written in standard form as ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. The solutions, or roots, are found using the quadratic formula x = [-b ± √(b² - 4ac)] / (2a). The graph of y = ax² + bx + c forms a parabola, which opens upward if a > 0 or downward if a < 0.
Key Properties of Quadratic Equations
Basic properties include the vertex, which is the parabola's turning point at x = -b/(2a), and the axis of symmetry, a vertical line through the vertex. The discriminant (b² - 4ac) determines the nature of roots: positive for two real roots, zero for one real root, and negative for no real roots. These properties help predict the parabola's shape, width, and position.
Practical Example: Projectile Motion
Consider a ball thrown upward with initial velocity 20 m/s from a height of 2 m. The height h(t) = -4.9t² + 20t + 2 models this as a quadratic equation, where t is time in seconds. The parabola peaks at the vertex (t ≈ 2.04 s, h ≈ 22.04 m), illustrating how quadratics capture acceleration due to gravity in real scenarios.
Real-World Applications and Importance
Quadratic equations model parabolas in diverse fields: projectile trajectories in sports and ballistics, profit maximization in business (e.g., revenue = -x² + 100x), and bridge arches in engineering. They are essential for optimization problems, enabling predictions and designs that account for curved paths, making abstract math practically applicable.