Math Algebra 2 Solutions

Quadratic Equations: Equation: x² + 3x + 2 = 0 a = 1, b = 3, c = 2 Axis of Symmetry: x = -3/2 Vertex: (-3/2, -1/4) Parabola Opens Up Min Value: -1/4 y-Intercept: (0, 2) Roots/Zeros: x = -2, x = -1 Domain: (-∞, ∞) Range: [-1/4, ∞) Equation: y = x² - 4x + 6 a = 1, b = -4, c = 6 Axis of Symmetry: x = 2 Vertex: (2, 2) Parabola Opens Up Min Value: 2 y-Intercept: (0, 6) Roots/Zeros: None Domain: (-∞, ∞) Range: [2, ∞) Transformations: f(x) = x² g(x) = -(x-3)² + 2 Transformations: Reflect over x-axis, Right 3, Up 2 Domain: (-∞, ∞) Range: (-∞, 2] f(x) = x² g(x) = 2(x+1)² - 5 Transformations: Vertical Stretch by 2, Left 1, Down 5 Domain: (-∞, ∞) Range: [-5, ∞) Factoring Problems: -2x³ + 16x y(y-6) + 9(y-6) -15x²y³ + 9xy⁴ 36f² - 16 25x⁴ - 64y⁴ (x+3)² - 4 n² - 7n + 6 2m² + m - 3 18y² - 21y - 30 2(a+b)² + 5(a+b) - 3 x³ + 4x² + x + 4 -9x³ - 3x² + 3x + 1 x² - 8x + 7 = 0 2p² + 12p = -10 -3x² - 3x + 9 = 0 Factoring Solutions: -2x(x² - 8) (y-6)(y+9) -3xy³(5x-3y) 4(9f² - 4) (5x² - 8y²)(5x² + 8y²) (x+1)(x+5) (n-6)(n-1) (2m+3)(m-1) 3(6y+5)(y-2) (2a+2b-1)(a+b+3) (x+4)(x²+1) (3x+1)(-3x²+1) (x-4)(x-3) (p+3)(p+2) (x+1/2)(x-3/2) Quadratic Equations and Solutions: Equation: x² + 7 = -8x x² + 8x + 7 = 0 x = -1, -7 Equation: 21x² - 8x = 3 21x² - 8x - 3 = 0 x = 1/3, 3/7 Equation: -x² = 49 -x² + 49 = 0 x = ±7 Equation: -8x² = -14x - 3 -8x² + 14x + 3 = 0 x = (7 ± √73) / 8 Story Problems: Problem: Mandy's age plus five times her age squared equals 36. Equation: m² + 5m = 36 Solution: m = 4 (ignoring the negative solution) Problem: Four times a number plus the square of the number equals 5. Equation: x² + 4x = 5 Solution: x = -5, 1 Problem 26: A rectangular flower bed is surrounded by a border of uniform width. The flower bed is 22 feet by 15 feet. The stone border will cover 120 square feet. Find the width of the border. Solution: Area of the whole garden = (15+2x)(22+2x) Area of the flower bed = 15*22 Area of the stones = 120 Equation: (15+2x)(22+2x) - 15*22 = 120 Solving the equation: 4x² + 74x - 120 = 0 (2x-3)(x+20) = 0 x = 3/2 or x = -20 (ignore the negative solution) The width of the border should be 3/2 feet. Problem 27: An internet service provider charges $30 per month to 1500 customers. For each $1 increase in price, the number of customers decreases by 25. Find the price that maximizes monthly revenue and the maximum revenue. Solution: Revenue = Cost * Quantity Revenue(x) = (30+x)(1500-25x) Revenue(x) = -25x² + 750x + 45000 This is a quadratic function with a downward-opening parabola. The vertex represents the maximum point. Axis of Symmetry: x = -750/(2*(-25)) = 15 Vertex: (15, 50625) Maximum Revenue: $50,625 The company should charge $45 per month to maximize revenue.