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.
ScanSolve AI Answer
1.2 Step 1: Identify integers n where n^2 < 17 < (n+1)^2. 4^2 = 16 5^2 = 25 Since 16 < 17 < 25, sqrt(17) lies between 4 and 5. 4 and 5 1.3.1 Step 1: Recognise perfect square trinomial x^2 - 2x + 1. Formula: (x - a)^2 = x^2 - 2ax + a^2. Here, 2a = 2 so a = 1. (x - 1)^2 = x^2 - 2x + 1. Thus, x^2 - 2x + 1 = (x - 1)^2. (x-1)^2 1.3.2 Step 1: Factor quadratic y^2 - y - 1. Discriminant: b^2 - 4ac = (-1)^2 - 4(1)(-1) = 1 + 4 = 5. Roots: y = 1 ± sqrt(5)2. Thus, y^2 - y - 1 = (y - 1 + sqrt(5)2)(y - 1 - sqrt(5)2). (y - 1 + sqrt(5)2)(y - 1 - sqrt(5)2) 1.3.3 Step 1: Rewrite 9x - x^2 = -(x^2 - 9x). Factor x^2 - 9x = x(x - 9). So -x(x - 9) = x(9 - x). Verify: x(9 - x) = 9x - x^2. x(9 - x) 1.3.4 Step 1: Recognise sum of cubes x^3 + 27 = x^3 + 3^3. Formula: a^3 + b^3 = (a + b)(a^2 - ab + b^2). Substitute a = x, b = 3: (x + 3)(x^2 - 3x + 9). Step 2: Verify expansion. x(x^2 - 3x + 9) = x^3 - 3x^2 + 9x 3(x^2 - 3x + 9) = 3x^2 - 9x + 27 Add: x^3 - 3x^2 + 9x + 3x^2 - 9x + 27 = x^3 + 27. (x + 3)(x^2 - 3x + 9) 1.4.1 Step 1: Expand (x - 3)(x^2 - 3x + 9). Distribute x: x · x^2 = x^3, x · (-3x) = -3x^2, x · 9 = 9x. Distribute -3: -3 · x^2 = -3x^2, -3 · (-3x) = 9x, -3 · 9 = -27. Step 2: Combine: x^3 + (-3x^2 - 3x^2) + (9x + 9x) - 27 = x^3 - 6x^2 + 18x - 27. x^3 - 6x^2 + 18x - 27 1.4.2 Step 1: Convert exponents. 25^-1 = (1)/(25), 5^-1 = (1)/(5). Expression: (1)/(25) - 1(1)/(5) + 1. Step 2: Numerator: (1 - 25)/(25) = -(24)/(25). Denominator: (1 + 5)/(5) = (6)/(5). Step 3: (-24/25)/(6/5) = -(24)/(25) × (5)/(6) = -(24 · 5)/(25 · 6) = -(120)/(150). Step 4: Simplify -(120 ÷ 30)/(150 ÷ 30) = -(4)/(5). -(4)/(5)