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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 369 · 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 = 14 and ab = 45, what is a² + b²? (a) 196 (b) 286 (c) 151 (d) 106
  2. 2.If a + b = 20 and ab = 99, what is a² + b²? (a) 202 (b) 598 (c) 400 (d) 301
  3. 3.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 961 (b) 971 (c) 901 (d) 841
  4. 4.What is the product of the roots of x² − 14x + 48 = 0? (a) 47 (b) 48 (c) 14 (d) 49
  5. 5.If a + b = 12 and ab = 35, what is a² + b²? (a) 109 (b) 74 (c) 214 (d) 144
  6. 6.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 14) (b) (x − 2)(x − 7) (c) (x + 1)(x + 8) (d) (x + 2)(x + 7)
  7. 7.If a + b = 11 and ab = 30, what is a² + b²? (a) 61 (b) 91 (c) 181 (d) 121
  8. 8.If a + b = 17 and ab = 66, what is (a − b)²? (a) 157 (b) 25 (c) 5 (d) 289
  9. 9.If a + b = 6 and ab = 8, what is (a − b)²? (a) 36 (b) 4 (c) 2 (d) 20
  10. 10.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 8)(x + 9) (c) (x − 8)(x − 9) (d) (x + 1)(x + 72)
  11. 11.If a + b = 16 and ab = 63, what is a² + b²? (a) 382 (b) 130 (c) 193 (d) 256
  12. 12.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8659 (b) 7569 (c) 8109 (d) 8649
  13. 13.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 36) (b) (x − 4)(x − 9) (c) (x + 1)(x + 12) (d) (x + 4)(x + 9)
  14. 14.If a + b = 16 and ab = 55, what is (a − b)²? (a) 36 (b) 256 (c) 146 (d) 6
  15. 15.What is the product of the roots of x² − 15x + 54 = 0? (a) 15 (b) 53 (c) 55 (d) 54
  16. 16.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 5476 (b) 4356 (c) 4916 (d) 5486
  17. 17.If a + b = 8 and ab = 15, what is a² + b²? (a) 64 (b) 49 (c) 34 (d) 94
  18. 18.What is the sum of the roots of x² − 13x + 42 = 0? (a) 12 (b) 13 (c) 42 (d) 14
  19. 19.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2401 (b) 2501 (c) 2601 (d) 2611
  20. 20.If a + b = 21 and ab = 108, what is (a − b)²? (a) 9 (b) 225 (c) 3 (d) 441
  21. 21.If a + b = 15 and ab = 56, what is a² + b²? (a) 113 (b) 169 (c) 337 (d) 225
  22. 22.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6251 (b) 6241 (c) 6399 (d) 6561
  23. 23.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 1575 (b) 2025 (c) 1225 (d) 1235
  24. 24.What is the product of the roots of x² − 12x + 35 = 0? (a) 36 (b) 34 (c) 35 (d) 12
  25. 25.If a + b = 19 and ab = 90, what is (a − b)²? (a) 361 (b) 181 (c) 1 (d) 271
  26. 26.If a + b = 10 and ab = 21, what is a² + b²? (a) 100 (b) 79 (c) 142 (d) 58
  27. 27.What are the roots of x² − 11x + 30 = 0? (a) 5 and 6 (b) 1 and 10 (c) 1 and 30 (d) 2 and 15
  28. 28.What is the sum of the roots of x² − 7x + 10 = 0? (a) 10 (b) 8 (c) 6 (d) 7
  29. 29.If a + b = 20 and ab = 96, what is a² + b²? (a) 400 (b) 592 (c) 208 (d) 304
  30. 30.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 2025 (b) 2525 (c) 3035 (d) 3025
  31. 31.What is the product of the roots of x² − 8x + 12 = 0? (a) 11 (b) 8 (c) 13 (d) 12
  32. 32.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 10609 (b) 9419 (c) 9991 (d) 9409
  33. 33.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4366 (b) 4356 (c) 4884 (d) 5476
  34. 34.If a + b = 5 and ab = 6, what is a² + b²? (a) 37 (b) 19 (c) 13 (d) 25
  35. 35.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 7) (b) (x + 1)(x + 12) (c) (x − 2)(x − 6) (d) (x + 2)(x + 6)
  36. 36.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x + 1)(x + 10) (c) (x + 5)(x + 6) (d) (x − 5)(x − 6)
  37. 37.If a + b = 17 and ab = 70, what is a² + b²? (a) 289 (b) 149 (c) 219 (d) 429
  38. 38.If a + b = 15 and ab = 54, what is (a − b)²? (a) 225 (b) 3 (c) 117 (d) 9
  39. 39.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 6) (b) (x + 1)(x + 10) (c) (x − 2)(x − 5) (d) (x + 2)(x + 5)
  40. 40.Which is the correct factorisation of x² + 10x + 16? (a) (x − 2)(x − 8) (b) (x + 1)(x + 16) (c) (x + 1)(x + 9) (d) (x + 2)(x + 8)

Answer key

  1. 106
  2. 202
  3. 961
  4. 48
  5. 74
  6. (x + 2)(x + 7)
  7. 61
  8. 25
  9. 4
  10. (x + 8)(x + 9)
  11. 130
  12. 8649
  13. (x + 4)(x + 9)
  14. 36
  15. 54
  16. 5476
  17. 34
  18. 13
  19. 2601
  20. 9
  21. 113
  22. 6241
  23. 1225
  24. 35
  25. 1
  26. 58
  27. 5 and 6
  28. 7
  29. 208
  30. 3025
  31. 12
  32. 9409
  33. 4356
  34. 13
  35. (x + 2)(x + 6)
  36. (x + 5)(x + 6)
  37. 149
  38. 9
  39. (x + 2)(x + 5)
  40. (x + 2)(x + 8)

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