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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 937 · 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 + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 15) (c) (x + 1)(x + 63) (d) (x + 7)(x + 9)
  2. 2.If a + b = 21 and ab = 108, what is a² + b²? (a) 333 (b) 441 (c) 225 (d) 657
  3. 3.If a + b = 9 and ab = 20, what is a² + b²? (a) 61 (b) 81 (c) 121 (d) 41
  4. 4.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 71 (c) 72 (d) 73
  5. 5.What are the roots of x² − 15x + 56 = 0? (a) 2 and 28 (b) 7 and 8 (c) 1 and 14 (d) 1 and 56
  6. 6.If a + b = 13 and ab = 42, what is (a − b)²? (a) 127 (b) 169 (c) 1 (d) 85
  7. 7.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x + 5)(x + 7) (c) (x + 1)(x + 35) (d) (x − 5)(x − 7)
  8. 8.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 299 (b) 391 (c) 529 (d) 289
  9. 9.What is the sum of the roots of x² − 12x + 27 = 0? (a) 11 (b) 27 (c) 13 (d) 12
  10. 10.What is the product of the roots of x² − 15x + 56 = 0? (a) 57 (b) 15 (c) 55 (d) 56
  11. 11.If a + b = 20 and ab = 96, what is a² + b²? (a) 400 (b) 208 (c) 304 (d) 592
  12. 12.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 1)(x + 14) (c) (x − 7)(x − 8) (d) (x + 7)(x + 8)
  13. 13.What is the product of the roots of x² − 15x + 54 = 0? (a) 53 (b) 15 (c) 54 (d) 55
  14. 14.If a + b = 18 and ab = 80, what is (a − b)²? (a) 2 (b) 4 (c) 324 (d) 164
  15. 15.If a + b = 12 and ab = 35, what is a² + b²? (a) 214 (b) 109 (c) 74 (d) 144
  16. 16.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3481 (b) 3491 (c) 3721 (d) 3599
  17. 17.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 289 (b) 409 (c) 539 (d) 529
  18. 18.If a + b = 13 and ab = 36, what is a² + b²? (a) 133 (b) 241 (c) 97 (d) 169
  19. 19.What is the sum of the roots of x² − 13x + 42 = 0? (a) 42 (b) 12 (c) 13 (d) 14
  20. 20.If a + b = 16 and ab = 63, what is a² + b²? (a) 193 (b) 130 (c) 256 (d) 382
  21. 21.If a + b = 19 and ab = 84, what is (a − b)²? (a) 361 (b) 25 (c) 193 (d) 5
  22. 22.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 7) (b) (x − 3)(x − 5) (c) (x + 3)(x + 5) (d) (x + 1)(x + 15)
  23. 23.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 904 (b) 784 (c) 1024 (d) 1034
  24. 24.If a + b = 14 and ab = 48, what is a² + b²? (a) 196 (b) 148 (c) 100 (d) 292
  25. 25.If a + b = 17 and ab = 72, what is a² + b²? (a) 289 (b) 145 (c) 433 (d) 217
  26. 26.If a + b = 14 and ab = 45, what is a² + b²? (a) 151 (b) 196 (c) 286 (d) 106
  27. 27.What is the product of the roots of x² − 16x + 63 = 0? (a) 16 (b) 63 (c) 62 (d) 64
  28. 28.If a + b = 15 and ab = 36, what is (a − b)²? (a) 153 (b) 81 (c) 9 (d) 225
  29. 29.What is the sum of the roots of x² − 13x + 40 = 0? (a) 14 (b) 40 (c) 13 (d) 12
  30. 30.If a + b = 16 and ab = 63, what is (a − b)²? (a) 256 (b) 2 (c) 130 (d) 4
  31. 31.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 7) (b) (x − 2)(x − 6) (c) (x + 1)(x + 12) (d) (x + 2)(x + 6)
  32. 32.If a + b = 19 and ab = 90, what is (a − b)²? (a) 181 (b) 361 (c) 271 (d) 1
  33. 33.If a + b = 5 and ab = 4, what is a² + b²? (a) 21 (b) 33 (c) 25 (d) 17
  34. 34.If a + b = 12 and ab = 32, what is (a − b)²? (a) 80 (b) 4 (c) 16 (d) 144
  35. 35.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1306 (b) 1936 (c) 1296 (d) 1584
  36. 36.If a + b = 19 and ab = 84, what is a² + b²? (a) 529 (b) 193 (c) 277 (d) 361
  37. 37.If a + b = 17 and ab = 66, what is a² + b²? (a) 223 (b) 421 (c) 157 (d) 289
  38. 38.If a + b = 11 and ab = 24, what is a² + b²? (a) 121 (b) 169 (c) 97 (d) 73
  39. 39.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 8091 (b) 8649 (c) 7579 (d) 7569
  40. 40.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 7056 (b) 7066 (c) 6416 (d) 5776

Answer key

  1. (x + 7)(x + 9)
  2. 225
  3. 41
  4. 72
  5. 7 and 8
  6. 1
  7. (x + 5)(x + 7)
  8. 289
  9. 12
  10. 56
  11. 208
  12. (x + 7)(x + 8)
  13. 54
  14. 4
  15. 74
  16. 3481
  17. 529
  18. 97
  19. 13
  20. 130
  21. 25
  22. (x + 3)(x + 5)
  23. 1024
  24. 100
  25. 145
  26. 106
  27. 63
  28. 81
  29. 13
  30. 4
  31. (x + 2)(x + 6)
  32. 1
  33. 17
  34. 16
  35. 1296
  36. 193
  37. 157
  38. 73
  39. 7569
  40. 7056

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