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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 119 · 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.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 24) (b) (x − 3)(x − 8) (c) (x + 1)(x + 10) (d) (x + 3)(x + 8)
  2. 2.If a + b = 17 and ab = 60, what is (a − b)²? (a) 169 (b) 7 (c) 289 (d) 49
  3. 3.If a + b = 20 and ab = 99, what is (a − b)²? (a) 400 (b) 4 (c) 202 (d) 2
  4. 4.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2916 (b) 2116 (c) 2484 (d) 2126
  5. 5.If a + b = 9 and ab = 14, what is a² + b²? (a) 109 (b) 81 (c) 67 (d) 53
  6. 6.If a + b = 15 and ab = 54, what is a² + b²? (a) 171 (b) 117 (c) 333 (d) 225
  7. 7.What is the product of the roots of x² − 14x + 48 = 0? (a) 49 (b) 47 (c) 48 (d) 14
  8. 8.If a + b = 11 and ab = 28, what is a² + b²? (a) 93 (b) 177 (c) 121 (d) 65
  9. 9.What is the product of the roots of x² − 7x + 12 = 0? (a) 11 (b) 12 (c) 7 (d) 13
  10. 10.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 9) (b) (x − 3)(x − 7) (c) (x + 1)(x + 21) (d) (x + 3)(x + 7)
  11. 11.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 4096 (b) 3136 (c) 3616 (d) 4106
  12. 12.If a + b = 10 and ab = 24, what is a² + b²? (a) 76 (b) 52 (c) 100 (d) 148
  13. 13.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 10) (b) (x − 2)(x − 5) (c) (x + 2)(x + 5) (d) (x + 1)(x + 6)
  14. 14.If a + b = 17 and ab = 70, what is (a − b)²? (a) 3 (b) 149 (c) 9 (d) 289
  15. 15.If a + b = 5 and ab = 6, what is a² + b²? (a) 19 (b) 37 (c) 13 (d) 25
  16. 16.What is the sum of the roots of x² − 10x + 21 = 0? (a) 21 (b) 9 (c) 10 (d) 11
  17. 17.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3844 (b) 3604 (c) 3364 (d) 3854
  18. 18.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 17 (c) 73 (d) 72
  19. 19.What are the roots of x² − 13x + 40 = 0? (a) 5 and 8 (b) 1 and 12 (c) 1 and 40 (d) 2 and 20
  20. 20.If a + b = 17 and ab = 72, what is a² + b²? (a) 217 (b) 145 (c) 289 (d) 433
  21. 21.If a + b = 8 and ab = 7, what is (a − b)²? (a) 36 (b) 50 (c) 64 (d) 6
  22. 22.If a + b = 10 and ab = 9, what is (a − b)²? (a) 100 (b) 82 (c) 64 (d) 8
  23. 23.If a + b = 13 and ab = 30, what is a² + b²? (a) 229 (b) 139 (c) 109 (d) 169
  24. 24.Which is the correct factorisation of x² + 5x + 6? (a) (x − 2)(x − 3) (b) (x + 1)(x + 6) (c) (x + 2)(x + 3) (d) (x + 1)(x + 4)
  25. 25.If a + b = 19 and ab = 84, what is a² + b²? (a) 361 (b) 193 (c) 277 (d) 529
  26. 26.What is the product of the roots of x² − 16x + 63 = 0? (a) 16 (b) 64 (c) 63 (d) 62
  27. 27.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3591 (b) 3969 (c) 3259 (d) 3249
  28. 28.What are the roots of x² − 16x + 63 = 0? (a) 1 and 15 (b) 7 and 9 (c) 1 and 63 (d) 3 and 21
  29. 29.If a + b = 21 and ab = 108, what is (a − b)²? (a) 225 (b) 3 (c) 9 (d) 441
  30. 30.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 324 (b) 494 (c) 484 (d) 404
  31. 31.What is the product of the roots of x² − 15x + 54 = 0? (a) 55 (b) 54 (c) 15 (d) 53
  32. 32.What is the sum of the roots of x² − 10x + 16 = 0? (a) 9 (b) 10 (c) 16 (d) 11
  33. 33.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8104 (b) 8474 (c) 7744 (d) 8464
  34. 34.If a + b = 6 and ab = 8, what is (a − b)²? (a) 4 (b) 2 (c) 36 (d) 20
  35. 35.What are the roots of x² − 17x + 72 = 0? (a) 8 and 9 (b) 1 and 72 (c) 1 and 16 (d) 2 and 36
  36. 36.If a + b = 12 and ab = 35, what is a² + b²? (a) 144 (b) 74 (c) 214 (d) 109
  37. 37.If a + b = 17 and ab = 70, what is a² + b²? (a) 289 (b) 149 (c) 429 (d) 219
  38. 38.If a + b = 16 and ab = 63, what is a² + b²? (a) 193 (b) 130 (c) 382 (d) 256
  39. 39.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 225 (b) 425 (c) 625 (d) 635
  40. 40.If a + b = 14 and ab = 40, what is (a − b)²? (a) 196 (b) 116 (c) 36 (d) 6

Answer key

  1. (x + 3)(x + 8)
  2. 49
  3. 4
  4. 2116
  5. 53
  6. 117
  7. 48
  8. 65
  9. 12
  10. (x + 3)(x + 7)
  11. 4096
  12. 52
  13. (x + 2)(x + 5)
  14. 9
  15. 13
  16. 10
  17. 3844
  18. 72
  19. 5 and 8
  20. 145
  21. 36
  22. 64
  23. 109
  24. (x + 2)(x + 3)
  25. 193
  26. 63
  27. 3249
  28. 7 and 9
  29. 9
  30. 484
  31. 54
  32. 10
  33. 8464
  34. 4
  35. 8 and 9
  36. 74
  37. 149
  38. 130
  39. 625
  40. 36

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