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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 329 · 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 59²? (a) 3721 (b) 3599 (c) 3481 (d) 3491
  2. 2.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x − 8)(x − 9) (c) (x + 1)(x + 16) (d) (x + 1)(x + 72)
  3. 3.What is the sum of the roots of x² − 8x + 15 = 0? (a) 15 (b) 7 (c) 8 (d) 9
  4. 4.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 7406 (b) 8084 (c) 8836 (d) 7396
  5. 5.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 539 (b) 289 (c) 409 (d) 529
  6. 6.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 225 (b) 375 (c) 625 (d) 235
  7. 7.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 8281 (b) 8099 (c) 7921 (d) 7931
  8. 8.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 1)(x + 13) (c) (x + 1)(x + 48) (d) (x + 6)(x + 8)
  9. 9.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x + 1)(x + 56) (c) (x + 1)(x + 14) (d) (x − 7)(x − 8)
  10. 10.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 6) (b) (x − 2)(x − 5) (c) (x + 2)(x + 5) (d) (x + 1)(x + 10)
  11. 11.If a + b = 10 and ab = 21, what is a² + b²? (a) 100 (b) 58 (c) 79 (d) 142
  12. 12.If a + b = 16 and ab = 63, what is (a − b)²? (a) 130 (b) 4 (c) 2 (d) 256
  13. 13.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6404 (b) 6084 (c) 6724 (d) 6734
  14. 14.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 4499 (b) 4891 (c) 4489 (d) 5329
  15. 15.If a + b = 20 and ab = 99, what is a² + b²? (a) 400 (b) 202 (c) 301 (d) 598
  16. 16.What are the roots of x² − 17x + 72 = 0? (a) 2 and 36 (b) 1 and 16 (c) 1 and 72 (d) 8 and 9
  17. 17.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8291 (b) 8101 (c) 7921 (d) 8281
  18. 18.If a + b = 14 and ab = 33, what is a² + b²? (a) 163 (b) 196 (c) 262 (d) 130
  19. 19.What is the sum of the roots of x² − 6x + 8 = 0? (a) 5 (b) 7 (c) 8 (d) 6
  20. 20.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 971 (b) 961 (c) 901 (d) 841
  21. 21.What is the sum of the roots of x² − 15x + 56 = 0? (a) 14 (b) 15 (c) 56 (d) 16
  22. 22.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 9226 (b) 10816 (c) 9984 (d) 9216
  23. 23.If a + b = 12 and ab = 35, what is a² + b²? (a) 214 (b) 144 (c) 74 (d) 109
  24. 24.If a + b = 9 and ab = 14, what is (a − b)²? (a) 81 (b) 5 (c) 53 (d) 25
  25. 25.What is the sum of the roots of x² − 11x + 18 = 0? (a) 10 (b) 11 (c) 12 (d) 18
  26. 26.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 4624 (b) 4896 (c) 4634 (d) 5184
  27. 27.If a + b = 21 and ab = 108, what is a² + b²? (a) 657 (b) 225 (c) 333 (d) 441
  28. 28.If a + b = 8 and ab = 15, what is a² + b²? (a) 94 (b) 49 (c) 64 (d) 34
  29. 29.If a + b = 11 and ab = 30, what is (a − b)²? (a) 121 (b) 61 (c) 91 (d) 1
  30. 30.Which is the correct factorisation of x² + 9x + 20? (a) (x + 4)(x + 5) (b) (x − 4)(x − 5) (c) (x + 1)(x + 20) (d) (x + 1)(x + 8)
  31. 31.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 1166 (b) 676 (c) 916 (d) 1156
  32. 32.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 16 (c) 17 (d) 72
  33. 33.If a + b = 15 and ab = 50, what is (a − b)²? (a) 5 (b) 25 (c) 225 (d) 125
  34. 34.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 904 (b) 784 (c) 1024 (d) 1034
  35. 35.What is the product of the roots of x² − 14x + 48 = 0? (a) 48 (b) 49 (c) 47 (d) 14
  36. 36.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1521 (b) 1599 (c) 1681 (d) 1531
  37. 37.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 8) (b) (x + 1)(x + 14) (c) (x − 2)(x − 7) (d) (x + 2)(x + 7)
  38. 38.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2209 (b) 2809 (c) 2219 (d) 2491
  39. 39.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 324 (b) 494 (c) 404 (d) 484
  40. 40.If a + b = 5 and ab = 6, what is (a − b)²? (a) 19 (b) 25 (c) 1 (d) 13

Answer key

  1. 3481
  2. (x + 8)(x + 9)
  3. 8
  4. 7396
  5. 529
  6. 225
  7. 7921
  8. (x + 6)(x + 8)
  9. (x + 7)(x + 8)
  10. (x + 2)(x + 5)
  11. 58
  12. 4
  13. 6724
  14. 4489
  15. 202
  16. 8 and 9
  17. 8281
  18. 130
  19. 6
  20. 961
  21. 15
  22. 9216
  23. 74
  24. 25
  25. 11
  26. 4624
  27. 225
  28. 34
  29. 1
  30. (x + 4)(x + 5)
  31. 1156
  32. 17
  33. 25
  34. 1024
  35. 48
  36. 1521
  37. (x + 2)(x + 7)
  38. 2209
  39. 484
  40. 1

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