{
  "title": "Solution by Completing the Square - Grade 10 CBSE",
  "total_questions": 100,
  "questions": [
    {"id": 1, "difficulty": "easy", "question": "To complete the square for x² + 6x, we add:", "options": {"A": "6", "B": "9", "C": "36", "D": "3"}, "correct_answer": "B"},
    {"id": 2, "difficulty": "easy", "question": "Solve x² + 4x - 5 = 0 by completing the square.", "options": {"A": "1, -5", "B": "-1, 5", "C": "2, -3", "D": "-2, 3"}, "correct_answer": "A"},
    {"id": 3, "difficulty": "easy", "question": "What should be added to x² - 8x to complete the square?", "options": {"A": "16", "B": "64", "C": "8", "D": "4"}, "correct_answer": "A"},
    {"id": 4, "difficulty": "easy", "question": "Solve x² - 2x - 3 = 0 by completing square.", "options": {"A": "3, -1", "B": "-3, 1", "C": "2, -2", "D": "1, -3"}, "correct_answer": "A"},
    {"id": 5, "difficulty": "easy", "question": "Complete square: x² + 10x + ___ = (x + ___)²", "options": {"A": "25, 5", "B": "100, 10", "C": "20, 10", "D": "5, 5"}, "correct_answer": "A"},
    {"id": 6, "difficulty": "easy", "question": "Solve x² + 8x + 15 = 0 by completing square.", "options": {"A": "-3, -5", "B": "3, 5", "C": "2, 6", "D": "-2, -6"}, "correct_answer": "A"},
    {"id": 7, "difficulty": "easy", "question": "What number completes square for x² - 12x?", "options": {"A": "36", "B": "144", "C": "24", "D": "6"}, "correct_answer": "A"},
    {"id": 8, "difficulty": "easy", "question": "Solve x² - 6x + 8 = 0 by completing square.", "options": {"A": "4, 2", "B": "-4, -2", "C": "3, 3", "D": "6, 1"}, "correct_answer": "A"},
    {"id": 9, "difficulty": "easy", "question": "Complete square: x² + 14x + ___ = (x + ___)²", "options": {"A": "49, 7", "B": "196, 14", "C": "28, 14", "D": "7, 7"}, "correct_answer": "A"},
    {"id": 10, "difficulty": "easy", "question": "Solve x² + 2x - 8 = 0 by completing square.", "options": {"A": "2, -4", "B": "-2, 4", "C": "1, -3", "D": "-1, 3"}, "correct_answer": "A"},
    {"id": 11, "difficulty": "easy", "question": "For x² + bx, term to add is:", "options": {"A": "(b/2)²", "B": "b²", "C": "(b)²/2", "D": "2b²"}, "correct_answer": "A"},
    {"id": 12, "difficulty": "easy", "question": "Solve x² - 10x + 21 = 0 by completing square.", "options": {"A": "7, 3", "B": "-7, -3", "C": "5, 5", "D": "10, 1"}, "correct_answer": "A"},
    {"id": 13, "difficulty": "easy", "question": "Complete square: x² - 16x + ___ = (x - ___)²", "options": {"A": "64, 8", "B": "256, 16", "C": "32, 16", "D": "8, 8"}, "correct_answer": "A"},
    {"id": 14, "difficulty": "easy", "question": "Solve x² + 12x + 32 = 0 by completing square.", "options": {"A": "-4, -8", "B": "4, 8", "C": "6, 6", "D": "-6, -6"}, "correct_answer": "A"},
    {"id": 15, "difficulty": "easy", "question": "What completes square for x² + 5x?", "options": {"A": "25/4", "B": "25", "C": "5/2", "D": "10"}, "correct_answer": "A"},
    {"id": 16, "difficulty": "easy", "question": "Solve x² - 4x - 12 = 0 by completing square.", "options": {"A": "6, -2", "B": "-6, 2", "C": "4, -3", "D": "-4, 3"}, "correct_answer": "A"},
    {"id": 17, "difficulty": "easy", "question": "Complete square: x² - 20x + ___ = (x - ___)²", "options": {"A": "100, 10", "B": "400, 20", "C": "40, 20", "D": "10, 10"}, "correct_answer": "A"},
    {"id": 18, "difficulty": "easy", "question": "Solve x² + 6x - 7 = 0 by completing square.", "options": {"A": "1, -7", "B": "-1, 7", "C": "3, -3", "D": "2, -4"}, "correct_answer": "A"},
    {"id": 19, "difficulty": "easy", "question": "For x² - 7x, add to complete square:", "options": {"A": "49/4", "B": "49", "C": "7/2", "D": "14"}, "correct_answer": "A"},
    {"id": 20, "difficulty": "easy", "question": "Solve x² - 8x + 7 = 0 by completing square.", "options": {"A": "7, 1", "B": "-7, -1", "C": "4, 2", "D": "8, -1"}, "correct_answer": "A"},

    {"id": 21, "difficulty": "medium", "question": "Solve 2x² - 7x + 3 = 0 by completing square.", "options": {"A": "3, 1/2", "B": "-3, -1/2", "C": "7/4, 1", "D": "2, 3/2"}, "correct_answer": "A"},
    {"id": 22, "difficulty": "medium", "question": "Complete square for 3x² + 6x - 2 = 0.", "options": {"A": "3(x+1)² - 5 = 0", "B": "3(x+1)² - 1 = 0", "C": "3(x+1)² - 3 = 0", "D": "3(x+1)² - 2 = 0"}, "correct_answer": "A"},
    {"id": 23, "difficulty": "medium", "question": "Solve 4x² - 12x + 5 = 0 by completing square.", "options": {"A": "5/2, 1/2", "B": "-5/2, -1/2", "C": "3/2, 1", "D": "2, 5/4"}, "correct_answer": "A"},
    {"id": 24, "difficulty": "medium", "question": "Solve x² + 3x - 10 = 0 by completing square.", "options": {"A": "2, -5", "B": "-2, 5", "C": "5/2, -4", "D": "3, -10/3"}, "correct_answer": "A"},
    {"id": 25, "difficulty": "medium", "question": "Complete square for x² + (1/2)x.", "options": {"A": "(x + 1/4)² - 1/16", "B": "(x + 1/2)² - 1/4", "C": "(x + 1/4)²", "D": "(x + 1/2)²"}, "correct_answer": "A"},
    {"id": 26, "difficulty": "medium", "question": "Solve 9x² - 15x + 6 = 0 by completing square.", "options": {"A": "2/3, 1", "B": "-2/3, -1", "C": "5/3, 2/3", "D": "1/3, 2"}, "correct_answer": "A"},
    {"id": 27, "difficulty": "medium", "question": "Complete square for 2x² - 5x + 2 = 0.", "options": {"A": "2(x - 5/4)² - 9/8 = 0", "B": "2(x - 5/4)² - 1/8 = 0", "C": "2(x - 5/4)² - 5/8 = 0", "D": "2(x - 5/4)² - 3/8 = 0"}, "correct_answer": "A"},
    {"id": 28, "difficulty": "medium", "question": "Solve x² - x - 6 = 0 by completing square.", "options": {"A": "3, -2", "B": "-3, 2", "C": "2, -3", "D": "1/2 ± √(25/4)"}, "correct_answer": "A"},
    {"id": 29, "difficulty": "medium", "question": "Complete square for x² + px.", "options": {"A": "(x + p/2)² - p²/4", "B": "(x + p)² - p²", "C": "(x + p/2)²", "D": "(x + p)²"}, "correct_answer": "A"},
    {"id": 30, "difficulty": "medium", "question": "Solve 16x² - 24x + 9 = 0 by completing square.", "options": {"A": "3/4, 3/4", "B": "-3/4, -3/4", "C": "4/3, 4/3", "D": "3/2, 3/2"}, "correct_answer": "A"},
    {"id": 31, "difficulty": "medium", "question": "Solve 5x² - 6x - 2 = 0 by completing square.", "options": {"A": "(3 ± √19)/5", "B": "(3 ± √29)/5", "C": "(6 ± √19)/5", "D": "(6 ± √29)/5"}, "correct_answer": "A"},
    {"id": 32, "difficulty": "medium", "question": "Complete square for ax² + bx + c = 0, a≠0.", "options": {"A": "a(x + b/2a)² = (b² - 4ac)/4a", "B": "a(x + b/2a)² = (4ac - b²)/4a", "C": "a(x + b/a)² = (b² - 4ac)/a", "D": "a(x + b/2a)² = b²/4a - c"}, "correct_answer": "A"},
    {"id": 33, "difficulty": "medium", "question": "Solve x² + 2√2x - 6 = 0 by completing square.", "options": {"A": "√2, -3√2", "B": "-√2, 3√2", "C": "√2 ± 2√2", "D": "-√2 ± 2√2"}, "correct_answer": "A"},
    {"id": 34, "difficulty": "medium", "question": "Complete square for 3x² + 4x - 5 = 0.", "options": {"A": "3(x + 2/3)² - 19/3 = 0", "B": "3(x + 2/3)² - 11/3 = 0", "C": "3(x + 2/3)² - 7/3 = 0", "D": "3(x + 2/3)² - 5/3 = 0"}, "correct_answer": "A"},
    {"id": 35, "difficulty": "medium", "question": "Solve 2x² + 5x + 2 = 0 by completing square.", "options": {"A": "-1/2, -2", "B": "1/2, 2", "C": "-2, 1/2", "D": "2, -1/2"}, "correct_answer": "A"},
    {"id": 36, "difficulty": "medium", "question": "Complete square for x² - (a+b)x + ab = 0.", "options": {"A": "[x - (a+b)/2]² = [(a-b)/2]²", "B": "[x - (a+b)/2]² = [(a+b)/2]²", "C": "[x - (a+b)]² = ab", "D": "[x - (a+b)/2]² = ab/4"}, "correct_answer": "A"},
    {"id": 37, "difficulty": "medium", "question": "Solve 4x² + 12x + 5 = 0 by completing square.", "options": {"A": "-1/2, -5/2", "B": "1/2, 5/2", "C": "-3/2, -5/2", "D": "3/2, 5/2"}, "correct_answer": "A"},
    {"id": 38, "difficulty": "medium", "question": "Complete square for x² + 2√3x + 2 = 0.", "options": {"A": "(x + √3)² = 1", "B": "(x + √3)² = 3", "C": "(x + √3)² = 2", "D": "(x + √3)² = 4"}, "correct_answer": "A"},
    {"id": 39, "difficulty": "medium", "question": "Solve 9x² + 6x - 8 = 0 by completing square.", "options": {"A": "2/3, -4/3", "B": "-2/3, 4/3", "C": "1/3, -8/3", "D": "-1/3, 8/3"}, "correct_answer": "A"},
    {"id": 40, "difficulty": "medium", "question": "Complete square for 2x² - 3x + 1 = 0.", "options": {"A": "2(x - 3/4)² - 1/8 = 0", "B": "2(x - 3/4)² + 1/8 = 0", "C": "2(x - 3/4)² - 9/8 = 0", "D": "2(x - 3/4)² - 3/8 = 0"}, "correct_answer": "A"},
   
  {"id": 41, "difficulty": "medium", "question": "Solve x² - 5x + 6 = 0 by completing square.", "options": {"A": "3, 2", "B": "-3, -2", "C": "5/2 ± 1/2", "D": "2, 3"}, "correct_answer": "D"},
  {"id": 42, "difficulty": "medium", "question": "Complete square for 4x² - 8x + 3 = 0.", "options": {"A": "4(x-1)² - 1 = 0", "B": "4(x-1)² + 1 = 0", "C": "4(x-1)² - 3 = 0", "D": "4(x-1)² - 2 = 0"}, "correct_answer": "A"},
  {"id": 43, "difficulty": "medium", "question": "Solve 25x² - 30x + 9 = 0 by completing square.", "options": {"A": "3/5, 3/5", "B": "-3/5, -3/5", "C": "5/3, 5/3", "D": "1/3, 3"}, "correct_answer": "A"},
  {"id": 44, "difficulty": "medium", "question": "Complete square for x² + (1/3)x - 2/3 = 0.", "options": {"A": "(x + 1/6)² = 25/36", "B": "(x + 1/6)² = 1/36", "C": "(x + 1/3)² = 4/9", "D": "(x + 1/3)² = 1/9"}, "correct_answer": "A"},
  {"id": 45, "difficulty": "medium", "question": "Solve 3x² - 2√6x + 2 = 0 by completing square.", "options": {"A": "√6/3, √6/3", "B": "-√6/3, -√6/3", "C": "√2, √3", "D": "2√6/3, √6/3"}, "correct_answer": "A"},
  {"id": 46, "difficulty": "medium", "question": "Complete square for 5x² + 2x - 3 = 0.", "options": {"A": "5(x + 1/5)² = 16/5", "B": "5(x + 1/5)² = 14/5", "C": "5(x + 1/5)² = 3", "D": "5(x + 1/5)² = 4"}, "correct_answer": "A"},
  {"id": 47, "difficulty": "medium", "question": "Solve x² - 3√2x + 4 = 0 by completing square.", "options": {"A": "2√2, √2", "B": "-2√2, -√2", "C": "√2 ± 2", "D": "3√2/2 ± 1/2"}, "correct_answer": "A"},
  {"id": 48, "difficulty": "medium", "question": "Complete square for ax² - bx + c = 0.", "options": {"A": "a(x - b/2a)² = (b² - 4ac)/4a", "B": "a(x - b/2a)² = (4ac - b²)/4a", "C": "a(x - b/a)² = (b² - 4ac)/a", "D": "a(x - b/2a)² = b²/4a - c"}, "correct_answer": "A"},
  {"id": 49, "difficulty": "medium", "question": "Solve 7x² - 10x + 3 = 0 by completing square.", "options": {"A": "1, 3/7", "B": "-1, -3/7", "C": "5/7 ± 2/7", "D": "10/7 ± 1/7"}, "correct_answer": "A"},
  {"id": 50, "difficulty": "medium", "question": "Complete square for x² + 2ax + b = 0.", "options": {"A": "(x + a)² = a² - b", "B": "(x + a)² = b - a²", "C": "(x + a)² = a² + b", "D": "(x + a)² = b² - a"}, "correct_answer": "A"},
  {"id": 51, "difficulty": "medium", "question": "Solve 6x² - 7x - 3 = 0 by completing square.", "options": {"A": "3/2, -1/3", "B": "-3/2, 1/3", "C": "7/12 ± √97/12", "D": "7/12 ± √85/12"}, "correct_answer": "A"},
  {"id": 52, "difficulty": "medium", "question": "Complete square for 2x² + 3x - 4 = 0.", "options": {"A": "2(x + 3/4)² = 41/8", "B": "2(x + 3/4)² = 25/8", "C": "2(x + 3/4)² = 17/8", "D": "2(x + 3/4)² = 9/8"}, "correct_answer": "A"},
  {"id": 53, "difficulty": "medium", "question": "Solve x² + 4√3x + 9 = 0 by completing square.", "options": {"A": "-√3, -3√3", "B": "√3, 3√3", "C": "-2√3 ± √3", "D": "-2√3 ± 3"}, "correct_answer": "A"},
  {"id": 54, "difficulty": "medium", "question": "Complete square for 9x² - 12x + 4 = 0.", "options": {"A": "9(x - 2/3)² = 0", "B": "9(x - 2/3)² = 4", "C": "9(x - 2/3)² = 8", "D": "9(x - 2/3)² = 12"}, "correct_answer": "A"},
  {"id": 55, "difficulty": "medium", "question": "Solve 8x² - 14x + 5 = 0 by completing square.", "options": {"A": "1, 5/8", "B": "-1, -5/8", "C": "7/8 ± 3/8", "D": "7/8 ± 1/8"}, "correct_answer": "A"},
  {"id": 56, "difficulty": "medium", "question": "Complete square for x² - px - q = 0.", "options": {"A": "(x - p/2)² = p²/4 + q", "B": "(x - p/2)² = p²/4 - q", "C": "(x - p/2)² = q - p²/4", "D": "(x - p)² = p² + q"}, "correct_answer": "A"},
  {"id": 57, "difficulty": "medium", "question": "Solve 10x² - 11x + 3 = 0 by completing square.", "options": {"A": "3/5, 1/2", "B": "-3/5, -1/2", "C": "11/20 ± 1/20", "D": "11/20 ± 7/20"}, "correct_answer": "A"},
  {"id": 58, "difficulty": "medium", "question": "Complete square for 4x² + 4√3x + 3 = 0.", "options": {"A": "4(x + √3/2)² = 0", "B": "4(x + √3/2)² = 1", "C": "4(x + √3/2)² = 2", "D": "4(x + √3/2)² = 3"}, "correct_answer": "A"},
  {"id": 59, "difficulty": "medium", "question": "Solve x² - (√5+1)x + √5 = 0 by completing square.", "options": {"A": "1, √5", "B": "-1, -√5", "C": "(√5+1)/2 ± (√5-1)/2", "D": "(√5+1)/2 ± √((√5-1)/2)"}, "correct_answer": "A"},
  {"id": 60, "difficulty": "medium", "question": "Complete square for ax² - 2bx + c = 0.", "options": {"A": "a(x - b/a)² = (b² - ac)/a", "B": "a(x - b/a)² = (ac - b²)/a", "C": "a(x - b/a)² = b²/a - c", "D": "a(x - b/a)² = (2b² - ac)/a"}, "correct_answer": "A"},
  {"id": 61, "difficulty": "medium", "question": "Solve 12x² - 13x + 3 = 0 by completing square.", "options": {"A": "3/4, 1/3", "B": "-3/4, -1/3", "C": "13/24 ± 5/24", "D": "13/24 ± 7/24"}, "correct_answer": "A"},
  {"id": 62, "difficulty": "medium", "question": "Complete square for x² + 2√2x - 6 = 0.", "options": {"A": "(x + √2)² = 8", "B": "(x + √2)² = 4", "C": "(x + √2)² = 6", "D": "(x + √2)² = 10"}, "correct_answer": "A"},
  {"id": 63, "difficulty": "medium", "question": "Solve 15x² - 16x + 4 = 0 by completing square.", "options": {"A": "2/3, 2/5", "B": "-2/3, -2/5", "C": "8/15 ± 2/15", "D": "8/15 ± 4/15"}, "correct_answer": "A"},
  {"id": 64, "difficulty": "medium", "question": "Complete square for 2x² - √2x - 2 = 0.", "options": {"A": "2(x - √2/4)² = 17/8", "B": "2(x - √2/4)² = 9/8", "C": "2(x - √2/4)² = 5/8", "D": "2(x - √2/4)² = 3/8"}, "correct_answer": "A"},
  {"id": 65, "difficulty": "medium", "question": "Solve x² - 3ax + 2a² = 0 by completing square.", "options": {"A": "2a, a", "B": "-2a, -a", "C": "(3a ± a)/2", "D": "a, 2a"}, "correct_answer": "D"},
  {"id": 66, "difficulty": "medium", "question": "Complete square for 3x² + 5x - 2 = 0.", "options": {"A": "3(x + 5/6)² = 49/12", "B": "3(x + 5/6)² = 25/12", "C": "3(x + 5/6)² = 13/12", "D": "3(x + 5/6)² = 7/12"}, "correct_answer": "A"},
  {"id": 67, "difficulty": "medium", "question": "Solve 4x² - 4√3x + 3 = 0 by completing square.", "options": {"A": "√3/2, √3/2", "B": "-√3/2, -√3/2", "C": "√3 ± 1/2", "D": "√3/2 ± 1/2"}, "correct_answer": "A"},
  {"id": 68, "difficulty": "medium", "question": "Complete square for x² - (a+b)x = -ab.", "options": {"A": "[x - (a+b)/2]² = [(a-b)/2]²", "B": "[x - (a+b)/2]² = [(a+b)/2]²", "C": "[x - (a+b)]² = ab", "D": "[x - (a+b)/2]² = ab/4"}, "correct_answer": "A"},
  {"id": 69, "difficulty": "medium", "question": "Solve 9x² + 18x + 8 = 0 by completing square.", "options": {"A": "-2/3, -4/3", "B": "2/3, 4/3", "C": "-1 ± 1/3", "D": "-1 ± 2/3"}, "correct_answer": "A"},
  {"id": 70, "difficulty": "medium", "question": "Complete square for px² + qx + r = 0, p≠0.", "options": {"A": "p(x + q/2p)² = (q² - 4pr)/4p", "B": "p(x + q/2p)² = (4pr - q²)/4p", "C": "p(x + q/p)² = (q² - 4pr)/p", "D": "p(x + q/2p)² = q²/4p - r"}, "correct_answer": "A"},
  {"id": 71, "difficulty": "medium", "question": "Solve 16x² - 40x + 25 = 0 by completing square.", "options": {"A": "5/4, 5/4", "B": "-5/4, -5/4", "C": "5/2, 5/2", "D": "5/8, 5/8"}, "correct_answer": "A"},
  {"id": 72, "difficulty": "medium", "question": "Complete square for 5x² - 3x + 1 = 0.", "options": {"A": "5(x - 3/10)² = -11/20", "B": "5(x - 3/10)² = 11/20", "C": "5(x - 3/10)² = -1/20", "D": "5(x - 3/10)² = 1/20"}, "correct_answer": "A"},
  {"id": 73, "difficulty": "medium", "question": "Solve x² + 2√5x + 5 = 0 by completing square.", "options": {"A": "-√5, -√5", "B": "√5, √5", "C": "-√5 ± √5", "D": "-√5 ± 0"}, "correct_answer": "A"},
  {"id": 74, "difficulty": "medium", "question": "Complete square for ax² + 2bx + c = 0.", "options": {"A": "a(x + b/a)² = (b² - ac)/a", "B": "a(x + b/a)² = (ac - b²)/a", "C": "a(x + b/a)² = b²/a - c", "D": "a(x + b/a)² = (2b² - ac)/a"}, "correct_answer": "A"},
  {"id": 75, "difficulty": "medium", "question": "Solve 25x² + 30x + 9 = 0 by completing square.", "options": {"A": "-3/5, -3/5", "B": "3/5, 3/5", "C": "-3/10, -3/10", "D": "3/10, 3/10"}, "correct_answer": "A"},
  {"id": 76, "difficulty": "medium", "question": "Complete square for x² - 2√3x + 2 = 0.", "options": {"A": "(x - √3)² = 1", "B": "(x - √3)² = 3", "C": "(x - √3)² = 2", "D": "(x - √3)² = 4"}, "correct_answer": "A"},
  {"id": 77, "difficulty": "medium", "question": "Solve 6x² + 11x + 4 = 0 by completing square.", "options": {"A": "-1/2, -4/3", "B": "1/2, 4/3", "C": "-11/12 ± 5/12", "D": "-11/12 ± 7/12"}, "correct_answer": "A"},
  {"id": 78, "difficulty": "medium", "question": "Complete square for 2x² + 4√2x + 3 = 0.", "options": {"A": "2(x + √2)² = 1", "B": "2(x + √2)² = 3", "C": "2(x + √2)² = 5", "D": "2(x + √2)² = 7"}, "correct_answer": "A"},
  {"id": 79, "difficulty": "medium", "question": "Solve x² - 4ax + 3a² = 0 by completing square.", "options": {"A": "3a, a", "B": "-3a, -a", "C": "(4a ± 2a)/2", "D": "a, 3a"}, "correct_answer": "D"},
  {"id": 80, "difficulty": "medium", "question": "Complete square for 4x² - 12x + 7 = 0.", "options": {"A": "4(x - 3/2)² = 2", "B": "4(x - 3/2)² = 4", "C": "4(x - 3/2)² = 6", "D": "4(x - 3/2)² = 8"}, "correct_answer": "A"},

  {"id": 81, "difficulty": "hard", "question": "Solve by completing square: 2√2x² + 7x + 5√2 = 0.", "options": {"A": "-√2, -5/√2", "B": "√2, 5/√2", "C": "-√2/2, -5√2/2", "D": "√2/2, 5√2/2"}, "correct_answer": "A"},
  {"id": 82, "difficulty": "hard", "question": "Complete square for: (a²+b²)x² - 2(ac+bd)x + (c²+d²) = 0.", "options": {"A": "(a²+b²)[x - (ac+bd)/(a²+b²)]² = [(ad-bc)/(a²+b²)]²", "B": "(a²+b²)[x - (ac+bd)/(a²+b²)]² = (ad-bc)²/(a²+b²)", "C": "(a²+b²)[x - (ac+bd)/(a²+b²)]² = 0 if ad=bc", "D": "All of these"}, "correct_answer": "D"},
  {"id": 83, "difficulty": "hard", "question": "Solve by completing square: x² - 2ax + a² - b² = 0.", "options": {"A": "a+b, a-b", "B": "-a-b, -a+b", "C": "a±b", "D": "b±a"}, "correct_answer": "C"},
  {"id": 84, "difficulty": "hard", "question": "Complete square for: 4x² - 4a²x + (a⁴ - b⁴) = 0.", "options": {"A": "4[x - a²/2]² = b⁴", "B": "4[x - a²/2]² = (a²-b²)²", "C": "4[x - a²/2]² = (a²+b²)²", "D": "4[x - a²/2]² = (a⁴-b⁴)"}, "correct_answer": "A"},
  {"id": 85, "difficulty": "hard", "question": "Solve by completing square: √3x² + 10x - 8√3 = 0.", "options": {"A": "2√3/3, -4√3", "B": "-2√3/3, 4√3", "C": "(-5 ± √73)/√3", "D": "(-5 ± √97)/√3"}, "correct_answer": "A"},
  {"id": 86, "difficulty": "hard", "question": "Complete square for: x² + 2(a+b)x + 4ab = 0.", "options": {"A": "[x + (a+b)]² = (a-b)²", "B": "[x + (a+b)]² = a²+b²", "C": "[x + (a+b)]² = 4ab", "D": "[x + (a+b)]² = (a+b)²-4ab"}, "correct_answer": "A"},
  {"id": 87, "difficulty": "hard", "question": "Solve by completing square: 9x² - 9(a+b)x + 2a²+5ab+2b² = 0.", "options": {"A": "(2a+b)/3, (a+2b)/3", "B": "-(2a+b)/3, -(a+2b)/3", "C": "(a+2b)/3, (2a+b)/3", "D": "(-a-2b)/3, (-2a-b)/3"}, "correct_answer": "A"},
  {"id": 88, "difficulty": "hard", "question": "Complete square for: (sin²A)x² - 2sinA·cosA·x + cos²A = 0.", "options": {"A": "sin²A(x - cotA)² = 0", "B": "sin²A(x - tanA)² = 0", "C": "sin²A(x - 1)² = cos²A", "D": "sin²A(x - cosA/sinA)² = 0"}, "correct_answer": "D"},
  {"id": 89, "difficulty": "hard", "question": "Solve by completing square: abx² + (b²-ac)x - bc = 0.", "options": {"A": "c/b, -b/a", "B": "-c/b, b/a", "C": "b/a, -c/b", "D": "-b/a, c/b"}, "correct_answer": "A"},
  {"id": 90, "difficulty": "hard", "question": "Complete square for: x² - 2x(a²+b²)/(a²-b²) + 1 = 0, a≠±b.", "options": {"A": "[x - (a²+b²)/(a²-b²)]² = [2ab/(a²-b²)]²", "B": "[x - (a²+b²)/(a²-b²)]² = [(a²-b²)/(2ab)]²", "C": "[x - (a²+b²)/(a²-b²)]² = 1", "D": "[x - (a²+b²)/(a²-b²)]² = 0"}, "correct_answer": "A"},
  {"id": 91, "difficulty": "hard", "question": "Solve by completing square: (a+b-2c)x² + (b+c-2a)x + (c+a-2b) = 0.", "options": {"A": "1, (c+a-2b)/(a+b-2c)", "B": "-1, (2b-c-a)/(a+b-2c)", "C": "1, (2b-c-a)/(a+b-2c)", "D": "-1, (c+a-2b)/(a+b-2c)"}, "correct_answer": "A"},
  {"id": 92, "difficulty": "hard", "question": "Complete square for: (p+q)²x² - 2(p²-q²)x + (p-q)² = 0.", "options": {"A": "(p+q)²[x - (p-q)/(p+q)]² = 0", "B": "(p+q)²[x - (p+q)/(p-q)]² = 0", "C": "(p+q)²[x - 1]² = (p-q)²", "D": "(p+q)²[x - (p²-q²)/(p+q)²]² = 0"}, "correct_answer": "A"},
  {"id": 93, "difficulty": "hard", "question": "Solve by completing square: (a-b)x² + (b-c)x + (c-a) = 0.", "options": {"A": "1, (c-a)/(a-b)", "B": "-1, (a-c)/(a-b)", "C": "1, (a-c)/(a-b)", "D": "-1, (c-a)/(a-b)"}, "correct_answer": "A"},
  {"id": 94, "difficulty": "hard", "question": "Complete square for: x² + 2(a-3)x + a²-6a+8 = 0.", "options": {"A": "[x + (a-3)]² = 1", "B": "[x + (a-3)]² = a²", "C": "[x + (a-3)]² = (a-4)(a-2)", "D": "[x + (a-3)]² = 4"}, "correct_answer": "A"},
  {"id": 95, "difficulty": "hard", "question": "Solve by completing square: √2x² + 7x + 5√2 = 0.", "options": {"A": "-√2, -5/√2", "B": "√2, 5/√2", "C": "-7/(2√2) ± 3/(2√2)", "D": "-7/(2√2) ± √89/(2√2)"}, "correct_answer": "A"},
  {"id": 96, "difficulty": "hard", "question": "Complete square for: (a²+b²+c²)x² - 2(ab+bc+ca)x + (a²+b²+c²) = 0.", "options": {"A": "(a²+b²+c²)[x - (ab+bc+ca)/(a²+b²+c²)]² = [(a²+b²+c²)² - (ab+bc+ca)²]/(a²+b²+c²)", "B": "(a²+b²+c²)[x - 1]² = 0", "C": "(a²+b²+c²)[x - (ab+bc+ca)/(a²+b²+c²)]² = 0 if a=b=c", "D": "Both A and C"}, "correct_answer": "D"},
  {"id": 97, "difficulty": "hard", "question": "Solve by completing square: x² - 2x(a cosθ + b sinθ) + a²+b² = 0.", "options": {"A": "a cosθ + b sinθ ± √(a²+b²) sinθ", "B": "a cosθ + b sinθ ± √(a²+b²) cosθ", "C": "a cosθ + b sinθ ± (a sinθ - b cosθ)", "D": "a cosθ + b sinθ ± √(a² sin²θ + b² cos²θ)"}, "correct_answer": "C"},
  {"id": 98, "difficulty": "hard", "question": "Complete square for: (a²-b²)x² - 2(ax-by) + 1 = 0.", "options": {"A": "Too complex expansion", "B": "Not a perfect square generally", "C": "Too complex, not a perfect square generally", "D": "Perfect square only if y = b/(a²-b²)"}, "correct_answer": "C"},
  {"id": 99, "difficulty": "hard", "question": "Solve by completing square: (x²+1)² = 4x(x²-1).", "options": {"A": "1, -1, √2, -√2", "B": "0, 1, -1", "C": "1, -1 only", "D": "No real roots"}, "correct_answer": "A"},
  {"id": 100, "difficulty": "hard", "question": "Complete square for the system: x²+y² = a², x+y = b. Then find condition for equal roots in resulting quadratic.", "options": {"A": "2b² = a²", "B": "b² = 2a²", "C": "b² = a²", "D": "2a² = b²"}, "correct_answer": "B"}
]
}