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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 316 · 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.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10001 (b) 10211 (c) 9801 (d) 10201
  2. 2.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 10816 (b) 9226 (c) 9216 (d) 9984
  3. 3.If a + b = 3 and ab = 2, what is a² + b²? (a) 9 (b) 13 (c) 7 (d) 5
  4. 4.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4366 (b) 5476 (c) 4884 (d) 4356
  5. 5.What are the roots of x² − 16x + 63 = 0? (a) 7 and 9 (b) 1 and 15 (c) 3 and 21 (d) 1 and 63
  6. 6.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 404 (b) 324 (c) 484 (d) 494
  7. 7.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 3146 (b) 3136 (c) 3584 (d) 4096
  8. 8.What is the product of the roots of x² − 9x + 20 = 0? (a) 19 (b) 21 (c) 20 (d) 9
  9. 9.If a + b = 19 and ab = 88, what is (a − b)²? (a) 361 (b) 185 (c) 9 (d) 3
  10. 10.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 73 (c) 71 (d) 72
  11. 11.What is the product of the roots of x² − 11x + 24 = 0? (a) 25 (b) 11 (c) 23 (d) 24
  12. 12.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 10) (b) (x + 3)(x + 8) (c) (x + 1)(x + 24) (d) (x − 3)(x − 8)
  13. 13.If a + b = 10 and ab = 9, what is a² + b²? (a) 82 (b) 118 (c) 91 (d) 100
  14. 14.What is the product of the roots of x² − 13x + 42 = 0? (a) 41 (b) 13 (c) 42 (d) 43
  15. 15.Which is the correct factorisation of x² + 11x + 28? (a) (x − 4)(x − 7) (b) (x + 1)(x + 10) (c) (x + 1)(x + 28) (d) (x + 4)(x + 7)
  16. 16.What are the roots of x² − 15x + 56 = 0? (a) 1 and 56 (b) 1 and 14 (c) 7 and 8 (d) 2 and 28
  17. 17.If a + b = 10 and ab = 21, what is a² + b²? (a) 100 (b) 142 (c) 79 (d) 58
  18. 18.If a + b = 13 and ab = 42, what is a² + b²? (a) 169 (b) 127 (c) 253 (d) 85
  19. 19.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6571 (b) 6401 (c) 6561 (d) 6241
  20. 20.If a + b = 17 and ab = 72, what is a² + b²? (a) 289 (b) 217 (c) 433 (d) 145
  21. 21.Which is the correct factorisation of x² + 11x + 18? (a) (x + 2)(x + 9) (b) (x + 1)(x + 10) (c) (x + 1)(x + 18) (d) (x − 2)(x − 9)
  22. 22.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 441 (b) 399 (c) 371 (d) 361
  23. 23.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3979 (b) 3969 (c) 3609 (d) 3249
  24. 24.What are the roots of x² − 10x + 21 = 0? (a) 3 and 7 (b) 2 and 8 (c) 1 and 21 (d) 1 and 9
  25. 25.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 7)(x + 8) (c) (x + 1)(x + 14) (d) (x + 1)(x + 56)
  26. 26.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 36) (b) (x + 1)(x + 12) (c) (x − 4)(x − 9) (d) (x + 4)(x + 9)
  27. 27.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8464 (b) 8104 (c) 7744 (d) 8474
  28. 28.What is the sum of the roots of x² − 10x + 16 = 0? (a) 16 (b) 11 (c) 9 (d) 10
  29. 29.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1849 (b) 1369 (c) 1591 (d) 1379
  30. 30.Which is the correct factorisation of x² + 10x + 21? (a) (x − 3)(x − 7) (b) (x + 1)(x + 9) (c) (x + 3)(x + 7) (d) (x + 1)(x + 21)
  31. 31.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2601 (b) 2401 (c) 2499 (d) 2411
  32. 32.If a + b = 17 and ab = 70, what is a² + b²? (a) 289 (b) 429 (c) 219 (d) 149
  33. 33.If a + b = 16 and ab = 48, what is a² + b²? (a) 256 (b) 208 (c) 352 (d) 160
  34. 34.What is the product of the roots of x² − 5x + 6 = 0? (a) 5 (b) 7 (c) 8 (d) 6
  35. 35.If a + b = 5 and ab = 6, what is a² + b²? (a) 25 (b) 13 (c) 37 (d) 19
  36. 36.If a + b = 9 and ab = 20, what is (a − b)²? (a) 81 (b) 61 (c) 1 (d) 41
  37. 37.If a + b = 18 and ab = 77, what is (a − b)²? (a) 4 (b) 170 (c) 16 (d) 324
  38. 38.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 5625 (b) 4225 (c) 4875 (d) 4235
  39. 39.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 4624 (b) 4904 (c) 5184 (d) 5194
  40. 40.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 1)(x + 72) (c) (x + 8)(x + 9) (d) (x − 8)(x − 9)

Answer key

  1. 10201
  2. 9216
  3. 5
  4. 4356
  5. 7 and 9
  6. 484
  7. 3136
  8. 20
  9. 9
  10. 72
  11. 24
  12. (x + 3)(x + 8)
  13. 82
  14. 42
  15. (x + 4)(x + 7)
  16. 7 and 8
  17. 58
  18. 85
  19. 6561
  20. 145
  21. (x + 2)(x + 9)
  22. 361
  23. 3969
  24. 3 and 7
  25. (x + 7)(x + 8)
  26. (x + 4)(x + 9)
  27. 8464
  28. 10
  29. 1369
  30. (x + 3)(x + 7)
  31. 2401
  32. 149
  33. 160
  34. 6
  35. 13
  36. 1
  37. 16
  38. 4225
  39. 5184
  40. (x + 8)(x + 9)

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