Understanding Quadratic Equations
A quadratic equation is a polynomial equation of degree two, typically written in the standard form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. To solve it, identify the values of a, b, and c, then apply methods like the quadratic formula, factoring, or completing the square to find the values of x that satisfy the equation.
Key Steps Using the Quadratic Formula
The quadratic formula is x = [-b ± √(b² - 4ac)] / (2a). First, ensure the equation is in standard form. Substitute a, b, and c into the formula. Calculate the discriminant D = b² - 4ac to determine the nature of roots: if D > 0, two real roots; D = 0, one real root; D < 0, two complex roots. Then compute the square root of D and solve for both positive and negative branches.
Practical Example
Consider the equation x² - 5x + 6 = 0. Here, a = 1, b = -5, c = 6. Discriminant D = (-5)² - 4(1)(6) = 25 - 24 = 1. Roots are x = [5 ± √1] / 2, so x = (5 + 1)/2 = 3 and x = (5 - 1)/2 = 2. Verification: (x - 3)(x - 2) = x² - 5x + 6, which matches.
Importance and Applications
Solving quadratic equations is essential in algebra and applies to physics for projectile motion, engineering for structural analysis, and economics for optimization problems. It models parabolic paths and helps predict outcomes in real-world scenarios involving acceleration or profit maximization.