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 the problems. Question 4: Solve for the unknown. 4.1.1 Step 1: Divide the equation by 2. 2x^2 - 14x + 12 = 0 x^2 - 7x + 6 = 0 Step 2: Factor the quadratic equation. (x-1)(x-6) = 0 Step 3: Solve for x. x-1 = 0 or x-6 = 0 x = 1 or x = 6 The values of x are 1 or 6. 4.1.2 Step 1: Rewrite (3-y) as -(y-3). y^2(y-3) + 81(3-y) = 0 y^2(y-3) - 81(y-3) = 0 Step 2: Factor out the common term (y-3). (y-3)(y^2 - 81) = 0 Step 3: Factor the difference of squares y^2 - 81. (y-3)(y-9)(y+9) = 0 Step 4: Solve for y. y-3 = 0 or y-9 = 0 or y+9 = 0 y = 3 or y = 9 or y = -9 The values of y are 3, 9, or -9. 4.1.3 Step 1: Find a common denominator and multiply the entire equation by it. The common denominator for 5 and 3 is 15. 15 ((x)/(5)) - 15 ((3x-2)/(3)) = 15(-1) 3x - 5(3x-2) = -15 Step 2: Distribute and simplify. 3x - 15x + 10 = -15 -12x + 10 = -15 Step 3: Isolate x. -12x = -15 - 10 -12x = -25 x = (-25)/(-12) x = (25)/(12) The value of x is (25)/(12). 4.1.4 Step 1: Define variables and set up equations based on the given information. Let M be the mother's current age and S be the son's current age. "A mother is currently nine times as old as her son": M = 9S (1) "In four years' time, she will be five times as old as her son": M+4 = 5(S+4) (2) Step 2: Substitute equation (1) into equation (2). 9S + 4 = 5(S+4) 9S + 4 = 5S + 20 Step 3: Solve for S. 9S - 5S = 20 - 4 4S = 16 S = (16)/(4) S = 4 Step 4: Substitute the value of S back into equation (1) to find M. M = 9(4) M = 36 The mother's current age is 36 years and the son's current age is 4 years. 4.1.5 Step 1: Express 243 as a power of 3. 3^2x = 243 3^2x = 3^5 Step 2: Equate the exponents. 2x = 5 Step 3: Solve for x. x = (5)/(2) The value of x is (5)/(2). 4.1.6 Step 1: Express (1)/(625) as a power of 5. 5^x+2 = (1)/(625) Since 625 = 5^4, we have: 5^x+2 = (1)/(5^4) 5^x+2 = 5^-4 Step 2: Equate the exponents. x+2 = -4 Step 3: Solve for x. x = -4 - 2 x = -6 The value of x is -6. Question 5 5.1.1 Step 1: Identify the pattern's common difference. The given pattern is 9; 15; 21; 27; ... The difference between consecutive terms is 15-9=6, 21-15=6, 27-21=6. The common difference is 6. Step 2: Calculate the next three terms. 27 + 6 = 33 33 + 6 = 39 39 + 6 = 45 The next three terms are 33, 39, 45. 5.1.2 The pattern is an arithmetic sequence where each term is obtained by adding 6 to the previous term. The first term is 9. 5.1.3 Step 1: Use the formula for the n^th term of an arithmetic sequence, T_n = a + (n-1)d. Here, the first term a=9, the common difference d=6, and we want the 20^th term, so n=20. Step 2: Substitute the values into the formula. T_20 = 9 + (20-1)6 T_20 = 9 + (19)6 T_20 = 9 + 114 T_20 = 123 The 20^th term of the pattern is 123. 5.2 Step 1: Identify the first term and the common difference. The given pattern is x+2; x+3; x+4; ... The first term a = x+2. The common difference d = (x+3) - (x+2) = 1. Step 2: Use the formula for the general rule (the n^th term) of an arithmetic sequence, T_n = a + (n-1)d. T_n = (x+2) + (n-1)(1) T_n = x+2+n-1 T_n = x+n+1 The general rule for the pattern is T_n = x+n+1. That's 2 down. 3 left today — send the next one.