Quadratic Equations — Solved Examples for Every Method

QUADRATIC EQUATIONS — SOLVED EXAMPLES Method 1: Factoring Example: Solve x² - 5x + 6 = 0 Step 1: Find two numbers that multiply to 6 and add to -5 → -2 and -3 Step 2: Rewrite as (x - 2)(x - 3) = 0 Step 3: Set each factor to zero → x = 2 or x = 3 Answer: x = 2, x = 3 Method 2: Completing the Square Example: Solve x² + 6x + 5 = 0 Step 1: Move constant → x² + 6x = -5 Step 2: Add (6/2)² = 9 to both sides → x² + 6x + 9 = 4 Step 3: Write as square → (x + 3)² = 4 Step 4: Take square root → x + 3 = ±2 Answer: x = -1 or x = -5 Method 3: Quadratic Formula Example: Solve 2x² + 3x - 2 = 0 Formula: x = [-b ± √(b² - 4ac)] / 2a Here a = 2, b = 3, c = -2 Step 1: Discriminant = 3² - 4(2)(-2) = 9 + 16 = 25 Step 2: x = [-3 ± √25] / 4 = [-3 ± 5] / 4 Answer: x = 1/2 or x = -2 PRACTICE QUESTIONS (for the student to try) 1. x² - 7x + 12 = 0 2. x² + 4x - 5 = 0 3. 3x² - 2x - 1 = 0