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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 532 · 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 61²? (a) 3481 (b) 3731 (c) 3601 (d) 3721
  2. 2.Which is the correct factorisation of x² + 14x + 45? (a) (x + 5)(x + 9) (b) (x − 5)(x − 9) (c) (x + 1)(x + 45) (d) (x + 1)(x + 13)
  3. 3.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 5776 (b) 7066 (c) 6416 (d) 7056
  4. 4.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2611 (b) 2601 (c) 2501 (d) 2401
  5. 5.Which is the correct factorisation of x² + 13x + 36? (a) (x − 4)(x − 9) (b) (x + 4)(x + 9) (c) (x + 1)(x + 12) (d) (x + 1)(x + 36)
  6. 6.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 1)(x + 16) (c) (x + 8)(x + 9) (d) (x − 8)(x − 9)
  7. 7.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
  8. 8.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 3625 (b) 4235 (c) 3025 (d) 4225
  9. 9.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 484 (b) 404 (c) 494 (d) 324
  10. 10.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 5) (b) (x − 2)(x − 4) (c) (x + 2)(x + 4) (d) (x + 1)(x + 8)
  11. 11.What are the roots of x² − 16x + 63 = 0? (a) 3 and 21 (b) 1 and 15 (c) 7 and 9 (d) 1 and 63
  12. 12.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 4225 (b) 3575 (c) 3025 (d) 3035
  13. 13.What are the roots of x² − 10x + 24 = 0? (a) 1 and 9 (b) 1 and 24 (c) 2 and 12 (d) 4 and 6
  14. 14.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)
  15. 15.What is the sum of the roots of x² − 6x + 8 = 0? (a) 7 (b) 6 (c) 5 (d) 8
  16. 16.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10609 (b) 9409 (c) 10619 (d) 10009
  17. 17.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 24) (b) (x − 4)(x − 6) (c) (x + 1)(x + 9) (d) (x + 4)(x + 6)
  18. 18.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 676 (b) 686 (c) 884 (d) 1156
  19. 19.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 909 (b) 1099 (c) 1089 (d) 729
  20. 20.What are the roots of x² − 13x + 42 = 0? (a) 1 and 42 (b) 6 and 7 (c) 1 and 12 (d) 2 and 21
  21. 21.If a + b = 18 and ab = 72, what is (a − b)²? (a) 324 (b) 180 (c) 6 (d) 36
  22. 22.Which is the correct factorisation of x² + 12x + 32? (a) (x − 4)(x − 8) (b) (x + 1)(x + 32) (c) (x + 1)(x + 11) (d) (x + 4)(x + 8)
  23. 23.If a + b = 14 and ab = 24, what is (a − b)²? (a) 100 (b) 196 (c) 10 (d) 148
  24. 24.What is the product of the roots of x² − 5x + 6 = 0? (a) 5 (b) 7 (c) 8 (d) 6
  25. 25.What are the roots of x² − 17x + 72 = 0? (a) 8 and 9 (b) 2 and 36 (c) 1 and 72 (d) 1 and 16
  26. 26.If a + b = 17 and ab = 70, what is a² + b²? (a) 219 (b) 149 (c) 429 (d) 289
  27. 27.If a + b = 21 and ab = 108, what is a² + b²? (a) 333 (b) 225 (c) 657 (d) 441
  28. 28.If a + b = 17 and ab = 66, what is (a − b)²? (a) 25 (b) 5 (c) 289 (d) 157
  29. 29.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 1034 (b) 1024 (c) 784 (d) 904
  30. 30.If a + b = 19 and ab = 90, what is (a − b)²? (a) 271 (b) 361 (c) 181 (d) 1
  31. 31.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2504 (b) 2704 (c) 2304 (d) 2714
  32. 32.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x + 1)(x + 14) (c) (x − 6)(x − 9) (d) (x + 6)(x + 9)
  33. 33.What is the product of the roots of x² − 7x + 12 = 0? (a) 11 (b) 12 (c) 7 (d) 13
  34. 34.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 16 (c) 72 (d) 17
  35. 35.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 7569 (b) 8109 (c) 8659 (d) 8649
  36. 36.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1596 (b) 1444 (c) 1764 (d) 1454
  37. 37.If a + b = 7 and ab = 10, what is a² + b²? (a) 39 (b) 69 (c) 29 (d) 49
  38. 38.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8281 (b) 8291 (c) 8101 (d) 7921
  39. 39.What is the product of the roots of x² − 11x + 18 = 0? (a) 19 (b) 18 (c) 11 (d) 17
  40. 40.If a + b = 11 and ab = 28, what is a² + b²? (a) 93 (b) 121 (c) 65 (d) 177

Answer key

  1. 3721
  2. (x + 5)(x + 9)
  3. 7056
  4. 2601
  5. (x + 4)(x + 9)
  6. (x + 8)(x + 9)
  7. 2 and 5
  8. 4225
  9. 484
  10. (x + 2)(x + 4)
  11. 7 and 9
  12. 3025
  13. 4 and 6
  14. (x + 2)(x + 8)
  15. 6
  16. 10609
  17. (x + 4)(x + 6)
  18. 676
  19. 1089
  20. 6 and 7
  21. 36
  22. (x + 4)(x + 8)
  23. 100
  24. 6
  25. 8 and 9
  26. 149
  27. 225
  28. 25
  29. 1024
  30. 1
  31. 2704
  32. (x + 6)(x + 9)
  33. 12
  34. 17
  35. 8649
  36. 1444
  37. 29
  38. 8281
  39. 18
  40. 65

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