Definition and Standard Form
Quadratic equations in algebra are second-degree polynomial equations of the form ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. This structure ensures the highest power of the variable is 2, distinguishing quadratics from linear or cubic equations. The properties revolve around their solutions (roots), graph, and behavior, providing a foundation for algebraic problem-solving.
Key Properties: Roots and Discriminant
The roots of a quadratic equation are the values of x that satisfy the equation, found using the quadratic formula x = [-b ± √(b² - 4ac)] / (2a). The discriminant D = b² - 4ac determines the nature of roots: if D > 0, two distinct real roots; D = 0, one real root (repeated); D < 0, two complex roots. Additionally, the sum of roots is -b/a and the product is c/a, enabling quick calculations without solving fully.
Graphical Properties: The Parabola
Graphically, quadratic equations represent parabolas. If a > 0, the parabola opens upward (minimum vertex); if a < 0, it opens downward (maximum vertex). The vertex form y = a(x - h)² + k highlights the vertex at (h, k), axis of symmetry x = h, and y-intercept at c. For example, graphing y = x² - 4x + 3 reveals roots at x=1 and x=3, with vertex at (2, -1).
Applications and Importance
Quadratic properties are crucial in fields like physics for projectile motion, economics for profit maximization, and engineering for optimization. They allow prediction of outcomes, such as maximum height in ball trajectories. Understanding these properties dispels misconceptions like assuming all quadratics have real roots, emphasizing the role of the discriminant in real-world applicability.