This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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You're on a roll — here are the solutions to the math problems: 21. a) Expand: (2x+3)(4-3x) Step 1: Use the distributive property (FOIL method). (2x+3)(4-3x) = 2x(4) + 2x(-3x) + 3(4) + 3(-3x) Step 2: Multiply the terms. = 8x - 6x^2 + 12 - 9x Step 3: Combine like terms. = -6x^2 + (8x - 9x) + 12 = -6x^2 - x + 12 The expanded expression is -6x^2 - x + 12. b) Factorize completely: x^2 - 7x + 10 Step 1: Find two numbers that multiply to 10 and add up to -7. The numbers are -2 and -5. Step 2: Rewrite the middle term using these numbers. x^2 - 2x - 5x + 10 Step 3: Factor by grouping. x(x-2) - 5(x-2) Step 4: Factor out the common binomial. (x-2)(x-5) The completely factorized expression is (x-2)(x-5). c) Solve the quadratic equation: x^2 - 5x - 6 = 0 Step 1: Find two numbers that multiply to -6 and add up to -5. The numbers are -6 and 1. Step 2: Rewrite the middle term using these numbers. x^2 - 6x + x - 6 = 0 Step 3: Factor by grouping. x(x-6) + 1(x-6) = 0 Step 4: Factor out the common binomial. (x+1)(x-6) = 0 Step 5: Set each factor equal to zero and solve for x. x+1 = 0 or x-6 = 0 x = -1 or x = 6 The solutions to the quadratic equation are x = -1 or x = 6. Got more? Send 'em.