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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Question 1(a) Step 1: Cross-multiply to eliminate fractions. 4(x^2 - 9) = 3(x^2 - 4) Step 2: Expand both sides. 4x^2 - 36 = 3x^2 - 12 Step 3: Subtract 3x^2 from both sides and add 36 to both sides. 4x^2 - 3x^2 - 36 + 36 = -12 + 36 x^2 = 24 Step 4: Take square root of both sides. x = ± sqrt(24) x = ± sqrt(4 · 6) = ± 2sqrt(6) Step 5: Verify solutions satisfy x ≠ ± 2. 2sqrt(6) ≈ 4.90 ≠ ± 2 and -2sqrt(6) ≈ -4.90 ≠ ± 2. Both valid. x = 2sqrt(6),\ -2sqrt(6) Question 1(b) Step 1: Multiply both sides by 2x(x + 3) to clear denominators (x ≠ 0, x ≠ -3). 2x(x + 3) ( (2x - 1)/(x + 3) - (3)/(2x) ) = 2x(x + 3) · (1)/(2) Step 2: Distribute on left side. First term: 2x(2x - 1) Second term: -3(x + 3) Right side: x(x + 3) 2x(2x - 1) - 3(x + 3) = x(x + 3) Step 3: Expand all terms. Left: 4x^2 - 2x - 3x - 9 = 4x^2 - 5x - 9 Right: x^2 + 3x 4x^2 - 5x - 9 = x^2 + 3x Step 4: Bring all terms to left side. 4x^2 - 5x - 9 - x^2 - 3x = 0 3x^2 - 8x - 9 = 0 Step 5: Solve using quadratic formula x = -b ± sqrt(b^2) - 4ac2a, where a=3, b=-8, c=-9. Discriminant: b^2 - 4ac = 64 - 4(3)(-9) = 64 + 108 = 172 x = 8 ± sqrt(172)6 = 8 ± 2sqrt(43)6 = 4 ± sqrt(43)3 Step 6: Verify solutions satisfy x ≠ 0, x ≠ -3. sqrt(43) ≈ 6.56, so x ≈ (4 + 6.56)/(3) ≈ 3.52 ≠ 0, -3 x ≈ (4 - 6.56)/(3) ≈ -0.85 ≠ 0, -3. Both valid. x = 4 + sqrt(43)3,\ 4 - sqrt(43)3