What Are Quadratic Equations?
Quadratic equations are polynomial equations of degree two, expressed in the standard form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. They represent parabolas when graphed and are fundamental in algebra for modeling relationships that involve squared variables.
Key Properties of Quadratic Equations
Quadratic equations have two roots, found using the quadratic formula x = [-b ± √(b² - 4ac)] / (2a), where the discriminant D = b² - 4ac determines the nature of roots: positive for two real roots, zero for one real root, and negative for complex roots. The vertex form y = a(x - h)² + k highlights the parabola's vertex at (h, k), axis of symmetry x = h, and direction based on a's sign.
Practical Example: Solving a Quadratic
Consider the equation x² - 5x + 6 = 0. Factoring gives (x - 2)(x - 3) = 0, so roots are x = 2 and x = 3. Graphically, it forms a parabola opening upwards with vertex at (2.5, -0.25). This example illustrates how quadratics model scenarios like projectile motion, where time to reach a height is calculated similarly.
Applications in Real-World Scenarios
Quadratic equations apply in physics for projectile trajectories, where height h(t) = -16t² + v₀t + h₀ models motion under gravity. In engineering, they optimize areas (e.g., maximizing fence-enclosed space) or costs. Economics uses them for profit maximization, and computer graphics for rendering curves, demonstrating their versatility beyond pure algebra.