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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 763 · 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 + 6x + 8? (a) (x + 2)(x + 4) (b) (x − 2)(x − 4) (c) (x + 1)(x + 5) (d) (x + 1)(x + 8)
  2. 2.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8291 (b) 8281 (c) 8101 (d) 7921
  3. 3.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 9801 (b) 10001 (c) 10211 (d) 10201
  4. 4.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 8096 (b) 7754 (c) 8464 (d) 7744
  5. 5.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x + 5)(x + 6) (c) (x − 5)(x − 6) (d) (x + 1)(x + 10)
  6. 6.If a + b = 17 and ab = 70, what is (a − b)²? (a) 149 (b) 289 (c) 3 (d) 9
  7. 7.What is the product of the roots of x² − 11x + 30 = 0? (a) 30 (b) 31 (c) 29 (d) 11
  8. 8.If a + b = 5 and ab = 6, what is a² + b²? (a) 13 (b) 37 (c) 25 (d) 19
  9. 9.What are the roots of x² − 11x + 30 = 0? (a) 1 and 30 (b) 1 and 10 (c) 5 and 6 (d) 2 and 15
  10. 10.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 289 (b) 409 (c) 529 (d) 539
  11. 11.What is the sum of the roots of x² − 16x + 63 = 0? (a) 63 (b) 15 (c) 16 (d) 17
  12. 12.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 16) (c) (x + 1)(x + 72) (d) (x + 8)(x + 9)
  13. 13.What is the product of the roots of x² − 14x + 45 = 0? (a) 14 (b) 46 (c) 44 (d) 45
  14. 14.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 4225 (b) 5625 (c) 5635 (d) 4925
  15. 15.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x − 7)(x − 8) (c) (x + 7)(x + 8) (d) (x + 1)(x + 56)
  16. 16.What are the roots of x² − 6x + 8 = 0? (a) 1 and 8 (b) 1 and 5 (c) 2 and 5 (d) 2 and 4
  17. 17.What are the roots of x² − 15x + 56 = 0? (a) 1 and 14 (b) 7 and 8 (c) 2 and 28 (d) 1 and 56
  18. 18.If a + b = 19 and ab = 84, what is (a − b)²? (a) 5 (b) 193 (c) 361 (d) 25
  19. 19.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6399 (b) 6241 (c) 6561 (d) 6251
  20. 20.If a + b = 16 and ab = 63, what is (a − b)²? (a) 130 (b) 256 (c) 4 (d) 2
  21. 21.If a + b = 4 and ab = 3, what is a² + b²? (a) 16 (b) 22 (c) 10 (d) 13
  22. 22.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 10016 (b) 9216 (c) 10816 (d) 10826
  23. 23.What is the sum of the roots of x² − 17x + 72 = 0? (a) 17 (b) 18 (c) 72 (d) 16
  24. 24.What are the roots of x² − 11x + 28 = 0? (a) 4 and 7 (b) 1 and 28 (c) 1 and 10 (d) 2 and 14
  25. 25.If a + b = 18 and ab = 80, what is a² + b²? (a) 244 (b) 324 (c) 164 (d) 484
  26. 26.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 625 (b) 225 (c) 425 (d) 635
  27. 27.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6084 (b) 6734 (c) 6404 (d) 6724
  28. 28.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 5476 (b) 4884 (c) 4366 (d) 4356
  29. 29.If a + b = 21 and ab = 108, what is (a − b)²? (a) 225 (b) 441 (c) 9 (d) 3
  30. 30.If a + b = 15 and ab = 36, what is (a − b)²? (a) 81 (b) 225 (c) 9 (d) 153
  31. 31.What are the roots of x² − 11x + 18 = 0? (a) 1 and 18 (b) 3 and 6 (c) 2 and 9 (d) 1 and 10
  32. 32.What is the sum of the roots of x² − 15x + 56 = 0? (a) 16 (b) 14 (c) 56 (d) 15
  33. 33.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 4624 (b) 5184 (c) 5194 (d) 4904
  34. 34.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1444 (b) 1604 (c) 1764 (d) 1774
  35. 35.If a + b = 19 and ab = 90, what is a² + b²? (a) 361 (b) 541 (c) 181 (d) 271
  36. 36.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 4106 (b) 3616 (c) 4096 (d) 3136
  37. 37.If a + b = 7 and ab = 6, what is a² + b²? (a) 43 (b) 61 (c) 37 (d) 49
  38. 38.If a + b = 6 and ab = 8, what is (a − b)²? (a) 4 (b) 20 (c) 36 (d) 2
  39. 39.If a + b = 7 and ab = 10, what is (a − b)²? (a) 9 (b) 49 (c) 29 (d) 3
  40. 40.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2501 (b) 2401 (c) 2611 (d) 2601

Answer key

  1. (x + 2)(x + 4)
  2. 8281
  3. 10201
  4. 7744
  5. (x + 5)(x + 6)
  6. 9
  7. 30
  8. 13
  9. 5 and 6
  10. 529
  11. 16
  12. (x + 8)(x + 9)
  13. 45
  14. 5625
  15. (x + 7)(x + 8)
  16. 2 and 4
  17. 7 and 8
  18. 25
  19. 6241
  20. 4
  21. 10
  22. 10816
  23. 17
  24. 4 and 7
  25. 164
  26. 625
  27. 6724
  28. 4356
  29. 9
  30. 81
  31. 2 and 9
  32. 15
  33. 5184
  34. 1764
  35. 181
  36. 4096
  37. 37
  38. 4
  39. 9
  40. 2601

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