Solution: 3x - 7 > 2x + 5 → 3x - 2x > 5 + 7 → x > 12
(12, ∞) Chapter 2: Algebraic Expressions 2.1 Key Formulas (Memorize!) | Identity | Expansion | |----------|-----------| | (a+b)² | a² + 2ab + b² | | (a-b)² | a² - 2ab + b² | | a² - b² | (a+b)(a-b) | | (a+b)³ | a³ + 3a²b + 3ab² + b³ | | (a-b)³ | a³ - 3a²b + 3ab² - b³ | | a³ + b³ | (a+b)(a² - ab + b²) | | a³ - b³ | (a-b)(a² + ab + b²) | 2.2 Worked Example Q: Factorize: x⁴ + x² + 1
Solution: Let opposite=3k, adjacent=4k Hypotenuse = √[(3k)² + (4k)²] = √(9k²+16k²) = 5k Class 9 Higher Math Solution Bd
sinθ = opposite/hyp = 3k/5k = 3/5 cosθ = adjacent/hyp = 4k/5k = 4/5 5.1 Quadratic Formula For ax² + bx + c = 0 (a≠0): x = [-b ± √(b² - 4ac)] / (2a)
Since AB = BC = 5, triangle ABC is isosceles. 4.1 Ratios for 0°, 30°, 45°, 60°, 90° | θ | sinθ | cosθ | tanθ | |---|------|------|------| | 0° | 0 | 1 | 0 | | 30° | 1/2 | √3/2 | 1/√3 | | 45° | √2/2 | √2/2 | 1 | | 60° | √3/2 | 1/2 | √3 | | 90° | 1 | 0 | ∞ | 4.2 Example Problem Q: If tanθ = 3/4, find sinθ and cosθ. Solution: 3x - 7 > 2x + 5
Solution: x⁴ + x² + 1 = (x⁴ + 2x² + 1) - x² = (x² + 1)² - (x)² = (x² + 1 - x)(x² + 1 + x) = (x² - x + 1)(x² + x + 1) 3.1 Distance Formula Distance between A(x₁, y₁) and B(x₂, y₂): AB = √[(x₂ - x₁)² + (y₂ - y₁)²] 3.2 Section Formula (Internal Division) Point P dividing AB internally in ratio m:n: P = ( (mx₂ + nx₁)/(m+n) , (my₂ + ny₁)/(m+n) ) 3.3 Worked Example Q: Show that points A(1,2), B(4,6), C(7,2) form an isosceles triangle.
x₁ = (5+3)/4 = 8/4 = 2 x₂ = (5-3)/4 = 2/4 = 1/2 x₁ = (5+3)/4 = 8/4 = 2 x₂
Solution: a=2, b=-5, c=2 Δ = (-5)² - 4×2×2 = 25 - 16 = 9 x = [5 ± √9] / (4) = [5 ± 3]/4
Solution: AB = √[(4-1)² + (6-2)²] = √(9+16) = √25 = 5 BC = √[(7-4)² + (2-6)²] = √(9+16) = 5 CA = √[(1-7)² + (2-2)²] = √(36+0) = 6