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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 665 · 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 = 18 and ab = 77, what is a² + b²? (a) 478 (b) 247 (c) 170 (d) 324
  2. 2.If a + b = 13 and ab = 36, what is a² + b²? (a) 169 (b) 133 (c) 97 (d) 241
  3. 3.What is the sum of the roots of x² − 9x + 20 = 0? (a) 8 (b) 20 (c) 9 (d) 10
  4. 4.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 7569 (b) 8649 (c) 8109 (d) 8659
  5. 5.If a + b = 21 and ab = 108, what is a² + b²? (a) 441 (b) 657 (c) 225 (d) 333
  6. 6.What is the sum of the roots of x² − 12x + 32 = 0? (a) 11 (b) 13 (c) 12 (d) 32
  7. 7.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x − 5)(x − 6) (c) (x + 1)(x + 10) (d) (x + 5)(x + 6)
  8. 8.If a + b = 18 and ab = 77, what is (a − b)²? (a) 324 (b) 170 (c) 16 (d) 4
  9. 9.Which is the correct factorisation of x² + 12x + 35? (a) (x − 5)(x − 7) (b) (x + 1)(x + 11) (c) (x + 1)(x + 35) (d) (x + 5)(x + 7)
  10. 10.What is the sum of the roots of x² − 12x + 35 = 0? (a) 11 (b) 35 (c) 13 (d) 12
  11. 11.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 8)(x + 9) (c) (x − 8)(x − 9) (d) (x + 1)(x + 16)
  12. 12.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 1034 (b) 1024 (c) 904 (d) 784
  13. 13.If a + b = 7 and ab = 10, what is (a − b)²? (a) 49 (b) 9 (c) 3 (d) 29
  14. 14.What is the sum of the roots of x² − 11x + 30 = 0? (a) 30 (b) 10 (c) 12 (d) 11
  15. 15.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 10) (b) (x + 1)(x + 28) (c) (x − 4)(x − 7) (d) (x + 4)(x + 7)
  16. 16.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x + 1)(x + 56) (c) (x − 7)(x − 8) (d) (x + 1)(x + 14)
  17. 17.If a + b = 17 and ab = 70, what is a² + b²? (a) 289 (b) 219 (c) 429 (d) 149
  18. 18.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3591 (b) 3259 (c) 3969 (d) 3249
  19. 19.What is the sum of the roots of x² − 11x + 18 = 0? (a) 12 (b) 18 (c) 11 (d) 10
  20. 20.If a + b = 9 and ab = 20, what is (a − b)²? (a) 61 (b) 41 (c) 1 (d) 81
  21. 21.What is the product of the roots of x² − 7x + 10 = 0? (a) 7 (b) 10 (c) 9 (d) 11
  22. 22.If a + b = 12 and ab = 27, what is a² + b²? (a) 117 (b) 198 (c) 90 (d) 144
  23. 23.If a + b = 19 and ab = 90, what is (a − b)²? (a) 181 (b) 361 (c) 1 (d) 271
  24. 24.If a + b = 11 and ab = 18, what is (a − b)²? (a) 7 (b) 85 (c) 49 (d) 121
  25. 25.What is the sum of the roots of x² − 17x + 72 = 0? (a) 17 (b) 16 (c) 18 (d) 72
  26. 26.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 24) (b) (x + 1)(x + 9) (c) (x + 4)(x + 6) (d) (x − 4)(x − 6)
  27. 27.What is the product of the roots of x² − 11x + 24 = 0? (a) 25 (b) 24 (c) 23 (d) 11
  28. 28.What is the sum of the roots of x² − 7x + 12 = 0? (a) 12 (b) 8 (c) 7 (d) 6
  29. 29.If a + b = 10 and ab = 24, what is (a − b)²? (a) 2 (b) 4 (c) 100 (d) 52
  30. 30.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 6416 (b) 5776 (c) 7056 (d) 7066
  31. 31.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6724 (b) 6094 (c) 6084 (d) 6396
  32. 32.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 909 (b) 1089 (c) 1099 (d) 729
  33. 33.If a + b = 9 and ab = 18, what is a² + b²? (a) 45 (b) 81 (c) 117 (d) 63
  34. 34.What is the product of the roots of x² − 15x + 56 = 0? (a) 57 (b) 56 (c) 55 (d) 15
  35. 35.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 2035 (b) 3025 (c) 2475 (d) 2025
  36. 36.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 401 (b) 361 (c) 451 (d) 441
  37. 37.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 8125 (b) 9025 (c) 9035 (d) 7225
  38. 38.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10619 (b) 10609 (c) 9409 (d) 10009
  39. 39.What is the sum of the roots of x² − 16x + 63 = 0? (a) 17 (b) 63 (c) 15 (d) 16
  40. 40.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 1166 (b) 676 (c) 1156 (d) 916

Answer key

  1. 170
  2. 97
  3. 9
  4. 8649
  5. 225
  6. 12
  7. (x + 5)(x + 6)
  8. 16
  9. (x + 5)(x + 7)
  10. 12
  11. (x + 8)(x + 9)
  12. 1024
  13. 9
  14. 11
  15. (x + 4)(x + 7)
  16. (x + 7)(x + 8)
  17. 149
  18. 3249
  19. 11
  20. 1
  21. 10
  22. 90
  23. 1
  24. 49
  25. 17
  26. (x + 4)(x + 6)
  27. 24
  28. 7
  29. 4
  30. 7056
  31. 6084
  32. 1089
  33. 45
  34. 56
  35. 2025
  36. 441
  37. 9025
  38. 10609
  39. 16
  40. 1156

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