Quadratic Equations

Answer

Step 1: Find two numbers that multiply to 6 and add to -5 → (-2) and (-3) Step 2: Factor: x² - 2x - 3x + 6 = x(x-2) - 3(x-2) = (x-2)(x-3) = 0 Step 3: Set each factor to zero: x-2 = 0 or x-3 = 0 Step 4

Answer

Step 1: Identify a=2, b=3, c=-2 Step 2: Calculate discriminant: b² - 4ac = 9 - 4(2)(-2) = 9 + 16 = 25 Step 3: Apply formula: x = [-3 ± √25] / 2(2) = [-3 ± 5] / 4 Step 4: Find roots: x = (-3+5)/4 = 1/2

Answer

Step 1: Move constant to RHS: x² + 6x = -5 Step 2: Complete square: add (6/2)² = 9 to both sides Step 3: x² + 6x + 9 = -5 + 9 = 4 Step 4: Factor LHS: (x + 3)² = 4 Step 5: Take square root: x + 3 = ±2

Answer

Step 1: Identify a=3, b=-4, c=2 Step 2: Calculate discriminant: Δ = b² - 4ac = (-4)² - 4(3)(2) = 16 - 24 = -8 Step 3: Analyze: Since Δ < 0, the equation has no real roots Step 4: Nature: Two complex c

Answer

Step 1: Add 9 to both sides: (x-2)² = 9 Step 2: Take square root of both sides: x-2 = ±3 Step 3: Solve both cases: Case 1: x-2 = 3 → x = 5 Case 2: x-2 = -3 → x = -1 Answer: x = 5, -1

Answer

Step 1: Let breadth = x metres, then length = (x+5) metres Step 2: Area equation: x(x+5) = 300 Step 3: Expand: x² + 5x = 300 Step 4: Rearrange: x² + 5x - 300 = 0 Step 5: Factor: (x+20)(x-15) = 0 Step

Answer

Use quadratic formula when: 1. Factorisation is difficult or not obvious 2. Coefficients are fractions or irrational numbers 3. Need exact answers quickly Example: x² + 3x - 1 = 0 (doesn't factor nic

Answer

Step 1: Find factors of 12 that add to -7: (-3) and (-4) Step 2: Split middle term: x² - 3x - 4x + 12 = 0 Step 3: Group terms: x(x-3) - 4(x-3) = 0 Step 4: Factor: (x-3)(x-4) = 0 Step 5: Solve: x-3 = 0