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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Here are the solutions to Question 3: 3.1 Simplify the following: 3.1.1 (2a-3)(2a+3) Step 1: Use the difference of squares formula, (A-B)(A+B) = A^2 - B^2. (2a-3)(2a+3) = (2a)^2 - (3)^2 Step 2: Simplify the terms. = 4a^2 - 9 The simplified expression is 4a^2 - 9. 3.1.2 (p^2-1)-2p(p-1) Step 1: Distribute the -2p into the second parenthesis. (p^2-1)-2p(p-1) = p^2-1 - (2p · p - 2p · 1) = p^2-1 - (2p^2-2p) Step 2: Remove the parenthesis and combine like terms. = p^2-1-2p^2+2p = (p^2-2p^2) + 2p - 1 = -p^2+2p-1 The simplified expression is -p^2+2p-1. 3.1.3 (4x^2y-8xy^3)/(xy) Step 1: Factor out the common term from the numerator. The common term is 4xy. (4x^2y-8xy^3)/(xy) = (4xy(x-2y^2))/(xy) Step 2: Cancel out the common term xy from the numerator and denominator. = 4(x-2y^2) = 4x-8y^2 The simplified expression is 4x-8y^2. 3.2 Factorise the following: 3.2.1 20x^2-45y^2 Step 1: Factor out the greatest common factor, which is 5. 20x^2-45y^2 = 5(4x^2-9y^2) Step 2: Recognize the difference of squares pattern, A^2-B^2 = (A-B)(A+B), where A=2x and B=3y. = 5((2x)^2-(3y)^2) = 5(2x-3y)(2x+3y) The factorised expression is 5(2x-3y)(2x+3y). 3.2.2 2x^2-5x-3 Step 1: Find two numbers that multiply to 2 × (-3) = -6 and add to -5. These numbers are -6 and 1. Step 2: Rewrite the middle term using these numbers and factor by grouping. 2x^2-5x-3 = 2x^2-6x+x-3 = 2x(x-3)+1(x-3) Step 3: Factor out the common binomial factor (x-3). = (x-3)(2x+1) The factorised expression is (x-3)(2x+1). 3.2.3 5x^2-15x+12y-4xy Step 1: Group the terms and factor out common factors from each group. (5x^2-15x) + (12y-4xy) = 5x(x-3) + 4y(3-x) Step 2: Notice that (3-x) is the negative of (x-3). Rewrite 4y(3-x) as -4y(x-3). = 5x(x-3) - 4y(x-3) Step 3: Factor out the common binomial factor (x-3). = (x-3)(5x-4y) The factorised expression is (x-3)(5x-4y). 3.3 Solve for x in each of the following equations: 3.3.1 3 · 5^x-21=54 Step 1: Add 21 to both sides of the equation. 3 · 5^x = 54+21 3 · 5^x = 75 Step 2: Divide both sides by 3. 5^x = (75)/(3) 5^x = 25 Step 3: Express 25 as a power of 5. 5^x = 5^2 Step 4: Equate the exponents. x=2 The solution is x=2. 3.3.2 2x^2-12x=14 Step 1: Move all terms to one side to form a standard quadratic equation. 2x^2-12x-14=0 Step 2: Divide the entire equation by 2 to simplify. x^2-6x-7=0 Step 3: Factor the quadratic equation. Find two numbers that multiply to -7 and add to -6. These numbers are -7 and 1. (x-7)(x+1)=0 Step 4: Set each factor equal to zero and solve for x. x-7=0 or x+1=0 x=7 or x=-1 The solutions are x=7 or x=-1. 3.4 Solve the following inequality and give your solution in interval notation: 2x-3 -10 Step 1: Add 3 to both sides of the inequality. 2x -10+3 2x -7 Step 2: Divide both sides by 2. x -(7)/(2) Step 3: Express the solution in interval notation. [-(7)/(2), ) The solution in interval notation is [-(7)/(2), ). 3.5 Solve algebraically for x and y in the following equations: 1) 5x-12=3y 2) 4x-5y=17 Step 1: Rewrite equation (1) in standard form Ax+By=C. 5x-3y=12 (Eq 1') Step 2: Use the elimination method. Multiply Eq 1' by 5 and Eq 2 by 3 to make the coefficients of y opposites. 5(5x-3y) = 5(12) 25x-15y=60 (Eq 3) 3(4x-5y) = 3(17) 12x-15y=51 (Eq 4) Step 3: Subtract Eq 4 from Eq 3 to eliminate y. (25x-15y) - (12x-15y) = 60-51 25x-12x = 9 13x = 9 Step 4: Solve for x. x = (9)/(13) Step 5: Substitute the value of x into Eq 1' (5x-3y=12) to solve for y. 5((9)/(13))-3y=12 (45)/(13)-3y=12 Step 6: Subtract (45)/(13) from both sides. -3y = 12 - (45)/(13) -3y = (12 × 13)/(13) - (45)/(13) -3y = (156-45)/(13) -3y = (111)/(13) Step 7: Divide by -3 to find y. y = (111)/(13 × (-3)) y = -(37)/(13) The solutions are x=(9)/(13), y=-(37)/(13). Send me the next one 📸