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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 512 · 40 questions
TalentJR

How to solve it (Use the identity, skip the guessing): (a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b²; so a² + b² = (a + b)² − 2ab and (a − b)² = (a + b)² − 4ab. For x² − Sx + P = 0: the sum of the roots is S and the product is P. To factorise x² + Sx + P, find two numbers that multiply to P and add up to S.

NameDateTime takenScore ___ / 40
  1. 1.What is the product of the roots of x² − 9x + 20 = 0? (a) 9 (b) 19 (c) 20 (d) 21
  2. 2.If a + b = 17 and ab = 70, what is (a − b)²? (a) 149 (b) 289 (c) 3 (d) 9
  3. 3.If a + b = 15 and ab = 44, what is a² + b²? (a) 313 (b) 225 (c) 137 (d) 181
  4. 4.Which is the correct factorisation of x² + 10x + 21? (a) (x − 3)(x − 7) (b) (x + 3)(x + 7) (c) (x + 1)(x + 21) (d) (x + 1)(x + 9)
  5. 5.What is the product of the roots of x² − 13x + 42 = 0? (a) 13 (b) 42 (c) 43 (d) 41
  6. 6.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2484 (b) 2126 (c) 2916 (d) 2116
  7. 7.If a + b = 10 and ab = 16, what is (a − b)²? (a) 68 (b) 100 (c) 36 (d) 6
  8. 8.If a + b = 13 and ab = 40, what is a² + b²? (a) 249 (b) 169 (c) 129 (d) 89
  9. 9.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 17 (c) 72 (d) 73
  10. 10.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 1225 (b) 635 (c) 625 (d) 875
  11. 11.If a + b = 21 and ab = 108, what is a² + b²? (a) 441 (b) 333 (c) 225 (d) 657
  12. 12.Which is the correct factorisation of x² + 15x + 54? (a) (x − 6)(x − 9) (b) (x + 6)(x + 9) (c) (x + 1)(x + 14) (d) (x + 1)(x + 54)
  13. 13.If a + b = 21 and ab = 108, what is (a − b)²? (a) 225 (b) 441 (c) 3 (d) 9
  14. 14.If a + b = 5 and ab = 4, what is (a − b)²? (a) 3 (b) 25 (c) 17 (d) 9
  15. 15.If a + b = 14 and ab = 24, what is (a − b)²? (a) 196 (b) 10 (c) 148 (d) 100
  16. 16.What are the roots of x² − 15x + 56 = 0? (a) 2 and 28 (b) 7 and 8 (c) 1 and 56 (d) 1 and 14
  17. 17.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 5) (b) (x − 2)(x − 4) (c) (x + 1)(x + 8) (d) (x + 2)(x + 4)
  18. 18.If a + b = 20 and ab = 99, what is (a − b)²? (a) 400 (b) 202 (c) 2 (d) 4
  19. 19.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 2025 (b) 3025 (c) 2525 (d) 3035
  20. 20.If a + b = 6 and ab = 8, what is a² + b²? (a) 20 (b) 28 (c) 36 (d) 52
  21. 21.If a + b = 9 and ab = 8, what is (a − b)²? (a) 49 (b) 7 (c) 81 (d) 65
  22. 22.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 27) (b) (x + 1)(x + 11) (c) (x − 3)(x − 9) (d) (x + 3)(x + 9)
  23. 23.What is the product of the roots of x² − 11x + 18 = 0? (a) 17 (b) 11 (c) 18 (d) 19
  24. 24.What is the sum of the roots of x² − 10x + 24 = 0? (a) 24 (b) 10 (c) 9 (d) 11
  25. 25.What is the sum of the roots of x² − 14x + 45 = 0? (a) 13 (b) 14 (c) 15 (d) 45
  26. 26.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 416 (b) 256 (c) 576 (d) 586
  27. 27.Which is the correct factorisation of x² + 12x + 35? (a) (x + 5)(x + 7) (b) (x + 1)(x + 11) (c) (x + 1)(x + 35) (d) (x − 5)(x − 7)
  28. 28.What are the roots of x² − 8x + 12 = 0? (a) 2 and 6 (b) 1 and 7 (c) 1 and 12 (d) 3 and 4
  29. 29.What is the sum of the roots of x² − 11x + 30 = 0? (a) 10 (b) 12 (c) 11 (d) 30
  30. 30.If a + b = 13 and ab = 40, what is (a − b)²? (a) 89 (b) 9 (c) 3 (d) 169
  31. 31.If a + b = 15 and ab = 56, what is (a − b)²? (a) 169 (b) 225 (c) 1 (d) 113
  32. 32.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 9999 (b) 10201 (c) 9811 (d) 9801
  33. 33.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1681 (b) 1601 (c) 1521 (d) 1691
  34. 34.If a + b = 17 and ab = 70, what is a² + b²? (a) 219 (b) 289 (c) 149 (d) 429
  35. 35.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 9) (b) (x + 1)(x + 24) (c) (x − 4)(x − 6) (d) (x + 4)(x + 6)
  36. 36.What is the sum of the roots of x² − 10x + 21 = 0? (a) 10 (b) 21 (c) 9 (d) 11
  37. 37.If a + b = 7 and ab = 12, what is a² + b²? (a) 73 (b) 49 (c) 37 (d) 25
  38. 38.If a + b = 10 and ab = 9, what is (a − b)²? (a) 100 (b) 8 (c) 64 (d) 82
  39. 39.If a + b = 10 and ab = 21, what is a² + b²? (a) 79 (b) 100 (c) 142 (d) 58
  40. 40.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 7921 (b) 8099 (c) 8281 (d) 7931

Answer key

  1. 20
  2. 9
  3. 137
  4. (x + 3)(x + 7)
  5. 42
  6. 2116
  7. 36
  8. 89
  9. 72
  10. 625
  11. 225
  12. (x + 6)(x + 9)
  13. 9
  14. 9
  15. 100
  16. 7 and 8
  17. (x + 2)(x + 4)
  18. 4
  19. 3025
  20. 20
  21. 49
  22. (x + 3)(x + 9)
  23. 18
  24. 10
  25. 14
  26. 576
  27. (x + 5)(x + 7)
  28. 2 and 6
  29. 11
  30. 9
  31. 1
  32. 9801
  33. 1681
  34. 149
  35. (x + 4)(x + 6)
  36. 10
  37. 25
  38. 64
  39. 58
  40. 7921

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