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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 289 · 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 74²? (a) 5476 (b) 4916 (c) 4356 (d) 5486
  2. 2.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 784 (b) 794 (c) 1024 (d) 896
  3. 3.What is the sum of the roots of x² − 11x + 30 = 0? (a) 30 (b) 11 (c) 12 (d) 10
  4. 4.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 35) (b) (x − 5)(x − 7) (c) (x + 5)(x + 7) (d) (x + 1)(x + 11)
  5. 5.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x − 7)(x − 9) (c) (x + 1)(x + 63) (d) (x + 7)(x + 9)
  6. 6.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x − 8)(x − 9) (c) (x + 1)(x + 72) (d) (x + 8)(x + 9)
  7. 7.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 5051 (b) 5041 (c) 4761 (d) 4901
  8. 8.If a + b = 19 and ab = 88, what is a² + b²? (a) 537 (b) 273 (c) 185 (d) 361
  9. 9.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 14) (b) (x + 1)(x + 54) (c) (x − 6)(x − 9) (d) (x + 6)(x + 9)
  10. 10.If a + b = 17 and ab = 70, what is a² + b²? (a) 149 (b) 289 (c) 219 (d) 429
  11. 11.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 9025 (b) 7225 (c) 8075 (d) 7235
  12. 12.If a + b = 21 and ab = 108, what is (a − b)²? (a) 441 (b) 225 (c) 3 (d) 9
  13. 13.Which is the correct factorisation of x² + 6x + 8? (a) (x + 2)(x + 4) (b) (x + 1)(x + 5) (c) (x − 2)(x − 4) (d) (x + 1)(x + 8)
  14. 14.If a + b = 16 and ab = 60, what is a² + b²? (a) 256 (b) 196 (c) 376 (d) 136
  15. 15.Which is the correct factorisation of x² + 7x + 12? (a) (x + 3)(x + 4) (b) (x + 1)(x + 12) (c) (x − 3)(x − 4) (d) (x + 1)(x + 6)
  16. 16.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)
  17. 17.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)
  18. 18.Which is the correct factorisation of x² + 8x + 12? (a) (x − 2)(x − 6) (b) (x + 2)(x + 6) (c) (x + 1)(x + 7) (d) (x + 1)(x + 12)
  19. 19.What is the sum of the roots of x² − 12x + 35 = 0? (a) 11 (b) 35 (c) 13 (d) 12
  20. 20.If a + b = 20 and ab = 99, what is (a − b)²? (a) 4 (b) 2 (c) 202 (d) 400
  21. 21.If a + b = 14 and ab = 45, what is a² + b²? (a) 106 (b) 151 (c) 286 (d) 196
  22. 22.If a + b = 15 and ab = 36, what is (a − b)²? (a) 81 (b) 9 (c) 225 (d) 153
  23. 23.If a + b = 5 and ab = 6, what is a² + b²? (a) 19 (b) 25 (c) 13 (d) 37
  24. 24.If a + b = 19 and ab = 90, what is a² + b²? (a) 361 (b) 541 (c) 181 (d) 271
  25. 25.What is the product of the roots of x² − 15x + 56 = 0? (a) 57 (b) 15 (c) 55 (d) 56
  26. 26.If a + b = 15 and ab = 44, what is (a − b)²? (a) 225 (b) 49 (c) 7 (d) 137
  27. 27.What is the sum of the roots of x² − 16x + 63 = 0? (a) 16 (b) 63 (c) 15 (d) 17
  28. 28.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1691 (b) 1681 (c) 1521 (d) 1601
  29. 29.Which is the correct factorisation of x² + 8x + 15? (a) (x + 3)(x + 5) (b) (x + 1)(x + 7) (c) (x + 1)(x + 15) (d) (x − 3)(x − 5)
  30. 30.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 7921 (b) 8281 (c) 8101 (d) 8291
  31. 31.If a + b = 10 and ab = 9, what is a² + b²? (a) 118 (b) 91 (c) 100 (d) 82
  32. 32.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3481 (b) 3599 (c) 3491 (d) 3721
  33. 33.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 45) (b) (x − 5)(x − 9) (c) (x + 5)(x + 9) (d) (x + 1)(x + 13)
  34. 34.If a + b = 18 and ab = 77, what is a² + b²? (a) 478 (b) 324 (c) 247 (d) 170
  35. 35.If a + b = 17 and ab = 72, what is a² + b²? (a) 289 (b) 433 (c) 145 (d) 217
  36. 36.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 8116 (b) 8846 (c) 7396 (d) 8836
  37. 37.If a + b = 3 and ab = 2, what is a² + b²? (a) 9 (b) 5 (c) 13 (d) 7
  38. 38.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 15 (c) 16 (d) 14
  39. 39.If a + b = 12 and ab = 35, what is (a − b)²? (a) 74 (b) 2 (c) 144 (d) 4
  40. 40.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x − 6)(x − 8) (c) (x + 6)(x + 8) (d) (x + 1)(x + 48)

Answer key

  1. 5476
  2. 784
  3. 11
  4. (x + 5)(x + 7)
  5. (x + 7)(x + 9)
  6. (x + 8)(x + 9)
  7. 5041
  8. 185
  9. (x + 6)(x + 9)
  10. 149
  11. 7225
  12. 9
  13. (x + 2)(x + 4)
  14. 136
  15. (x + 3)(x + 4)
  16. (x + 4)(x + 9)
  17. (x + 3)(x + 9)
  18. (x + 2)(x + 6)
  19. 12
  20. 4
  21. 106
  22. 81
  23. 13
  24. 181
  25. 56
  26. 49
  27. 16
  28. 1681
  29. (x + 3)(x + 5)
  30. 8281
  31. 82
  32. 3481
  33. (x + 5)(x + 9)
  34. 170
  35. 145
  36. 8836
  37. 5
  38. 15
  39. 4
  40. (x + 6)(x + 8)

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