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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 657 · 40 questions
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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 64²? (a) 4096 (b) 3616 (c) 3136 (d) 4106
  2. 2.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 12) (b) (x + 2)(x + 6) (c) (x + 1)(x + 7) (d) (x − 2)(x − 6)
  3. 3.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6571 (b) 6561 (c) 6401 (d) 6241
  4. 4.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 399 (b) 441 (c) 361 (d) 371
  5. 5.What is the product of the roots of x² − 10x + 21 = 0? (a) 10 (b) 20 (c) 21 (d) 22
  6. 6.Which is the correct factorisation of x² + 15x + 54? (a) (x − 6)(x − 9) (b) (x + 1)(x + 14) (c) (x + 6)(x + 9) (d) (x + 1)(x + 54)
  7. 7.What is the sum of the roots of x² − 11x + 24 = 0? (a) 10 (b) 24 (c) 12 (d) 11
  8. 8.If a + b = 17 and ab = 72, what is (a − b)²? (a) 145 (b) 217 (c) 1 (d) 289
  9. 9.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 7)(x + 8) (c) (x − 7)(x − 8) (d) (x + 1)(x + 14)
  10. 10.If a + b = 19 and ab = 88, what is (a − b)²? (a) 9 (b) 361 (c) 185 (d) 3
  11. 11.If a + b = 11 and ab = 10, what is (a − b)²? (a) 101 (b) 121 (c) 81 (d) 9
  12. 12.What is the sum of the roots of x² − 10x + 16 = 0? (a) 10 (b) 11 (c) 9 (d) 16
  13. 13.What is the sum of the roots of x² − 12x + 27 = 0? (a) 13 (b) 27 (c) 11 (d) 12
  14. 14.If a + b = 17 and ab = 70, what is a² + b²? (a) 429 (b) 149 (c) 289 (d) 219
  15. 15.If a + b = 10 and ab = 24, what is a² + b²? (a) 76 (b) 148 (c) 52 (d) 100
  16. 16.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 5184 (b) 4904 (c) 5194 (d) 4624
  17. 17.What is the sum of the roots of x² − 11x + 18 = 0? (a) 10 (b) 12 (c) 11 (d) 18
  18. 18.What is the product of the roots of x² − 15x + 56 = 0? (a) 56 (b) 55 (c) 15 (d) 57
  19. 19.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 7056 (b) 7066 (c) 6416 (d) 5776
  20. 20.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 15) (c) (x + 1)(x + 63) (d) (x + 7)(x + 9)
  21. 21.If a + b = 19 and ab = 90, what is a² + b²? (a) 271 (b) 181 (c) 541 (d) 361
  22. 22.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 2025 (b) 2035 (c) 3025 (d) 2475
  23. 23.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 1156 (b) 676 (c) 686 (d) 884
  24. 24.If a + b = 16 and ab = 48, what is a² + b²? (a) 256 (b) 352 (c) 208 (d) 160
  25. 25.Which is the correct factorisation of x² + 7x + 10? (a) (x − 2)(x − 5) (b) (x + 2)(x + 5) (c) (x + 1)(x + 6) (d) (x + 1)(x + 10)
  26. 26.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 925 (b) 625 (c) 1235 (d) 1225
  27. 27.If a + b = 12 and ab = 35, what is a² + b²? (a) 109 (b) 74 (c) 144 (d) 214
  28. 28.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 909 (b) 729 (c) 1089 (d) 1099
  29. 29.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 1)(x + 42) (c) (x + 1)(x + 12) (d) (x + 6)(x + 7)
  30. 30.Which is the correct factorisation of x² + 12x + 32? (a) (x + 4)(x + 8) (b) (x + 1)(x + 11) (c) (x + 1)(x + 32) (d) (x − 4)(x − 8)
  31. 31.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8659 (b) 8109 (c) 8649 (d) 7569
  32. 32.What is the sum of the roots of x² − 9x + 14 = 0? (a) 8 (b) 14 (c) 10 (d) 9
  33. 33.If a + b = 15 and ab = 56, what is (a − b)²? (a) 113 (b) 1 (c) 169 (d) 225
  34. 34.If a + b = 18 and ab = 72, what is a² + b²? (a) 252 (b) 180 (c) 324 (d) 468
  35. 35.If a + b = 11 and ab = 28, what is a² + b²? (a) 121 (b) 65 (c) 177 (d) 93
  36. 36.Which is the correct factorisation of x² + 14x + 45? (a) (x − 5)(x − 9) (b) (x + 1)(x + 13) (c) (x + 5)(x + 9) (d) (x + 1)(x + 45)
  37. 37.If a + b = 17 and ab = 72, what is a² + b²? (a) 289 (b) 217 (c) 433 (d) 145
  38. 38.If a + b = 17 and ab = 60, what is (a − b)²? (a) 169 (b) 7 (c) 289 (d) 49
  39. 39.If a + b = 5 and ab = 4, what is a² + b²? (a) 25 (b) 17 (c) 33 (d) 21
  40. 40.What is the sum of the roots of x² − 16x + 63 = 0? (a) 17 (b) 63 (c) 15 (d) 16

Answer key

  1. 4096
  2. (x + 2)(x + 6)
  3. 6561
  4. 361
  5. 21
  6. (x + 6)(x + 9)
  7. 11
  8. 1
  9. (x + 7)(x + 8)
  10. 9
  11. 81
  12. 10
  13. 12
  14. 149
  15. 52
  16. 5184
  17. 11
  18. 56
  19. 7056
  20. (x + 7)(x + 9)
  21. 181
  22. 2025
  23. 676
  24. 160
  25. (x + 2)(x + 5)
  26. 1225
  27. 74
  28. 1089
  29. (x + 6)(x + 7)
  30. (x + 4)(x + 8)
  31. 8649
  32. 9
  33. 1
  34. 180
  35. 65
  36. (x + 5)(x + 9)
  37. 145
  38. 49
  39. 17
  40. 16

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