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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 373 · 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.If a + b = 19 and ab = 90, what is (a − b)²? (a) 1 (b) 361 (c) 181 (d) 271
  2. 2.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 7225 (b) 9035 (c) 9025 (d) 8125
  3. 3.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 1024 (b) 784 (c) 896 (d) 794
  4. 4.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 1156 (b) 676 (c) 686 (d) 884
  5. 5.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 18 (c) 17 (d) 16
  6. 6.What is the sum of the roots of x² − 12x + 35 = 0? (a) 13 (b) 11 (c) 12 (d) 35
  7. 7.What are the roots of x² − 14x + 45 = 0? (a) 1 and 13 (b) 1 and 45 (c) 3 and 15 (d) 5 and 9
  8. 8.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2704 (b) 2714 (c) 2504 (d) 2304
  9. 9.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 12) (b) (x − 2)(x − 6) (c) (x + 2)(x + 6) (d) (x + 1)(x + 7)
  10. 10.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 4624 (b) 4904 (c) 5194 (d) 5184
  11. 11.If a + b = 5 and ab = 4, what is a² + b²? (a) 17 (b) 25 (c) 33 (d) 21
  12. 12.If a + b = 17 and ab = 66, what is (a − b)²? (a) 157 (b) 5 (c) 25 (d) 289
  13. 13.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 529 (b) 299 (c) 391 (d) 289
  14. 14.Which is the correct factorisation of x² + 15x + 54? (a) (x + 6)(x + 9) (b) (x + 1)(x + 14) (c) (x + 1)(x + 54) (d) (x − 6)(x − 9)
  15. 15.If a + b = 16 and ab = 48, what is a² + b²? (a) 256 (b) 208 (c) 352 (d) 160
  16. 16.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 15) (b) (x + 1)(x + 7) (c) (x − 3)(x − 5) (d) (x + 3)(x + 5)
  17. 17.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 3616 (b) 3136 (c) 4096 (d) 4106
  18. 18.If a + b = 15 and ab = 50, what is (a − b)²? (a) 225 (b) 125 (c) 5 (d) 25
  19. 19.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 1 and 16 (c) 8 and 9 (d) 2 and 36
  20. 20.If a + b = 15 and ab = 36, what is (a − b)²? (a) 153 (b) 225 (c) 9 (d) 81
  21. 21.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2601 (b) 2401 (c) 2499 (d) 2411
  22. 22.If a + b = 21 and ab = 108, what is a² + b²? (a) 225 (b) 333 (c) 441 (d) 657
  23. 23.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3604 (b) 3854 (c) 3364 (d) 3844
  24. 24.If a + b = 15 and ab = 54, what is a² + b²? (a) 117 (b) 333 (c) 225 (d) 171
  25. 25.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1764 (b) 1774 (c) 1444 (d) 1604
  26. 26.If a + b = 5 and ab = 6, what is a² + b²? (a) 19 (b) 25 (c) 37 (d) 13
  27. 27.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 4096 (b) 3584 (c) 3136 (d) 3146
  28. 28.If a + b = 12 and ab = 27, what is (a − b)²? (a) 36 (b) 6 (c) 90 (d) 144
  29. 29.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3481 (b) 3721 (c) 3599 (d) 3491
  30. 30.If a + b = 9 and ab = 8, what is a² + b²? (a) 81 (b) 73 (c) 97 (d) 65
  31. 31.If a + b = 7 and ab = 12, what is (a − b)²? (a) 25 (b) 37 (c) 49 (d) 1
  32. 32.If a + b = 8 and ab = 15, what is (a − b)²? (a) 64 (b) 4 (c) 34 (d) 2
  33. 33.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 904 (b) 1034 (c) 784 (d) 1024
  34. 34.If a + b = 9 and ab = 18, what is (a − b)²? (a) 9 (b) 3 (c) 81 (d) 45
  35. 35.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 9) (b) (x + 1)(x + 16) (c) (x − 2)(x − 8) (d) (x + 2)(x + 8)
  36. 36.If a + b = 9 and ab = 18, what is a² + b²? (a) 81 (b) 63 (c) 117 (d) 45
  37. 37.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 875 (b) 635 (c) 625 (d) 1225
  38. 38.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 7235 (b) 5625 (c) 6425 (d) 7225
  39. 39.What is the sum of the roots of x² − 15x + 54 = 0? (a) 16 (b) 54 (c) 14 (d) 15
  40. 40.What are the roots of x² − 15x + 56 = 0? (a) 1 and 14 (b) 2 and 28 (c) 7 and 8 (d) 1 and 56

Answer key

  1. 1
  2. 9025
  3. 784
  4. 676
  5. 17
  6. 12
  7. 5 and 9
  8. 2704
  9. (x + 2)(x + 6)
  10. 5184
  11. 17
  12. 25
  13. 289
  14. (x + 6)(x + 9)
  15. 160
  16. (x + 3)(x + 5)
  17. 4096
  18. 25
  19. 8 and 9
  20. 81
  21. 2401
  22. 225
  23. 3844
  24. 117
  25. 1764
  26. 13
  27. 3136
  28. 36
  29. 3481
  30. 65
  31. 1
  32. 4
  33. 1024
  34. 9
  35. (x + 2)(x + 8)
  36. 45
  37. 625
  38. 7225
  39. 15
  40. 7 and 8

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