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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 635 · 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 72²? (a) 4904 (b) 4624 (c) 5194 (d) 5184
  2. 2.If a + b = 12 and ab = 11, what is a² + b²? (a) 144 (b) 122 (c) 133 (d) 166
  3. 3.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 8) (b) (x − 2)(x − 4) (c) (x + 1)(x + 5) (d) (x + 2)(x + 4)
  4. 4.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8281 (b) 8291 (c) 7921 (d) 8101
  5. 5.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 7066 (b) 5776 (c) 6416 (d) 7056
  6. 6.If a + b = 19 and ab = 90, what is (a − b)²? (a) 271 (b) 361 (c) 181 (d) 1
  7. 7.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 3136 (b) 3146 (c) 4096 (d) 3584
  8. 8.What are the roots of x² − 13x + 36 = 0? (a) 1 and 36 (b) 2 and 18 (c) 4 and 9 (d) 1 and 12
  9. 9.Which is the correct factorisation of x² + 12x + 27? (a) (x − 3)(x − 9) (b) (x + 1)(x + 11) (c) (x + 1)(x + 27) (d) (x + 3)(x + 9)
  10. 10.If a + b = 19 and ab = 84, what is (a − b)²? (a) 193 (b) 25 (c) 5 (d) 361
  11. 11.What are the roots of x² − 13x + 42 = 0? (a) 1 and 12 (b) 2 and 21 (c) 6 and 7 (d) 1 and 42
  12. 12.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 576 (b) 256 (c) 586 (d) 416
  13. 13.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)
  14. 14.Which is the correct factorisation of x² + 5x + 6? (a) (x + 2)(x + 3) (b) (x − 2)(x − 3) (c) (x + 1)(x + 6) (d) (x + 1)(x + 4)
  15. 15.If a + b = 20 and ab = 99, what is (a − b)²? (a) 400 (b) 202 (c) 2 (d) 4
  16. 16.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x + 1)(x + 72) (c) (x + 1)(x + 16) (d) (x − 8)(x − 9)
  17. 17.What are the roots of x² − 12x + 35 = 0? (a) 1 and 35 (b) 1 and 11 (c) 2 and 10 (d) 5 and 7
  18. 18.If a + b = 14 and ab = 24, what is (a − b)²? (a) 100 (b) 196 (c) 10 (d) 148
  19. 19.What are the roots of x² − 7x + 10 = 0? (a) 1 and 10 (b) 3 and 4 (c) 1 and 6 (d) 2 and 5
  20. 20.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1454 (b) 1444 (c) 1764 (d) 1596
  21. 21.If a + b = 9 and ab = 18, what is a² + b²? (a) 45 (b) 81 (c) 117 (d) 63
  22. 22.If a + b = 5 and ab = 6, what is (a − b)²? (a) 13 (b) 1 (c) 25 (d) 19
  23. 23.Which is the correct factorisation of x² + 9x + 18? (a) (x − 3)(x − 6) (b) (x + 1)(x + 18) (c) (x + 1)(x + 8) (d) (x + 3)(x + 6)
  24. 24.If a + b = 19 and ab = 88, what is a² + b²? (a) 273 (b) 537 (c) 185 (d) 361
  25. 25.What is the product of the roots of x² − 15x + 54 = 0? (a) 15 (b) 53 (c) 55 (d) 54
  26. 26.If a + b = 13 and ab = 36, what is (a − b)²? (a) 169 (b) 97 (c) 25 (d) 5
  27. 27.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2601 (b) 2501 (c) 2611 (d) 2401
  28. 28.If a + b = 19 and ab = 88, what is (a − b)²? (a) 361 (b) 185 (c) 9 (d) 3
  29. 29.What are the roots of x² − 12x + 32 = 0? (a) 2 and 16 (b) 1 and 32 (c) 4 and 8 (d) 1 and 11
  30. 30.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 18 (c) 17 (d) 72
  31. 31.If a + b = 17 and ab = 66, what is (a − b)²? (a) 289 (b) 157 (c) 5 (d) 25
  32. 32.If a + b = 15 and ab = 54, what is (a − b)²? (a) 117 (b) 3 (c) 225 (d) 9
  33. 33.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2314 (b) 2704 (c) 2496 (d) 2304
  34. 34.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 361 (b) 441 (c) 371 (d) 399
  35. 35.What is the sum of the roots of x² − 15x + 56 = 0? (a) 16 (b) 56 (c) 15 (d) 14
  36. 36.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 14) (b) (x + 2)(x + 7) (c) (x − 2)(x − 7) (d) (x + 1)(x + 8)
  37. 37.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 4489 (b) 4891 (c) 5329 (d) 4499
  38. 38.If a + b = 21 and ab = 108, what is (a − b)²? (a) 441 (b) 225 (c) 9 (d) 3
  39. 39.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 5635 (b) 6375 (c) 7225 (d) 5625
  40. 40.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2401 (b) 2499 (c) 2601 (d) 2411

Answer key

  1. 5184
  2. 122
  3. (x + 2)(x + 4)
  4. 8281
  5. 7056
  6. 1
  7. 3136
  8. 4 and 9
  9. (x + 3)(x + 9)
  10. 25
  11. 6 and 7
  12. 576
  13. (x + 3)(x + 8)
  14. (x + 2)(x + 3)
  15. 4
  16. (x + 8)(x + 9)
  17. 5 and 7
  18. 100
  19. 2 and 5
  20. 1444
  21. 45
  22. 1
  23. (x + 3)(x + 6)
  24. 185
  25. 54
  26. 25
  27. 2601
  28. 9
  29. 4 and 8
  30. 17
  31. 25
  32. 9
  33. 2304
  34. 361
  35. 15
  36. (x + 2)(x + 7)
  37. 4489
  38. 9
  39. 5625
  40. 2401

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