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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 586 · 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 99²? (a) 9811 (b) 10201 (c) 9999 (d) 9801
  2. 2.If a + b = 19 and ab = 90, what is (a − b)²? (a) 361 (b) 181 (c) 1 (d) 271
  3. 3.If a + b = 18 and ab = 72, what is a² + b²? (a) 324 (b) 468 (c) 252 (d) 180
  4. 4.If a + b = 6 and ab = 8, what is a² + b²? (a) 52 (b) 36 (c) 20 (d) 28
  5. 5.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 73 (c) 71 (d) 72
  6. 6.What is the product of the roots of x² − 14x + 45 = 0? (a) 14 (b) 44 (c) 46 (d) 45
  7. 7.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 4106 (b) 3616 (c) 3136 (d) 4096
  8. 8.If a + b = 17 and ab = 72, what is (a − b)²? (a) 1 (b) 217 (c) 289 (d) 145
  9. 9.What is the product of the roots of x² − 15x + 56 = 0? (a) 15 (b) 55 (c) 56 (d) 57
  10. 10.What is the sum of the roots of x² − 13x + 36 = 0? (a) 14 (b) 13 (c) 12 (d) 36
  11. 11.If a + b = 12 and ab = 11, what is a² + b²? (a) 144 (b) 133 (c) 166 (d) 122
  12. 12.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1584 (b) 1936 (c) 1296 (d) 1306
  13. 13.What are the roots of x² − 11x + 30 = 0? (a) 1 and 30 (b) 5 and 6 (c) 2 and 15 (d) 1 and 10
  14. 14.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 384 (b) 576 (c) 256 (d) 266
  15. 15.If a + b = 8 and ab = 15, what is (a − b)²? (a) 34 (b) 2 (c) 4 (d) 64
  16. 16.What is the sum of the roots of x² − 7x + 12 = 0? (a) 6 (b) 8 (c) 7 (d) 12
  17. 17.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)
  18. 18.If a + b = 15 and ab = 54, what is a² + b²? (a) 333 (b) 171 (c) 117 (d) 225
  19. 19.What is the sum of the roots of x² − 5x + 6 = 0? (a) 6 (b) 4 (c) 5 (d) 7
  20. 20.What are the roots of x² − 15x + 56 = 0? (a) 1 and 56 (b) 2 and 28 (c) 1 and 14 (d) 7 and 8
  21. 21.Which is the correct factorisation of x² + 12x + 32? (a) (x − 4)(x − 8) (b) (x + 4)(x + 8) (c) (x + 1)(x + 11) (d) (x + 1)(x + 32)
  22. 22.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x + 7)(x + 8) (c) (x + 1)(x + 56) (d) (x − 7)(x − 8)
  23. 23.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 5339 (b) 4489 (c) 5329 (d) 4909
  24. 24.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 18 (c) 17 (d) 16
  25. 25.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 11035 (b) 11025 (c) 10025 (d) 9025
  26. 26.If a + b = 12 and ab = 35, what is (a − b)²? (a) 74 (b) 4 (c) 144 (d) 2
  27. 27.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 441 (b) 401 (c) 451 (d) 361
  28. 28.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 72) (c) (x + 1)(x + 16) (d) (x + 8)(x + 9)
  29. 29.If a + b = 17 and ab = 60, what is (a − b)²? (a) 7 (b) 169 (c) 49 (d) 289
  30. 30.What are the roots of x² − 12x + 35 = 0? (a) 5 and 7 (b) 2 and 10 (c) 1 and 11 (d) 1 and 35
  31. 31.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 28) (b) (x − 4)(x − 7) (c) (x + 4)(x + 7) (d) (x + 1)(x + 10)
  32. 32.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 4356 (b) 5476 (c) 5486 (d) 4916
  33. 33.If a + b = 13 and ab = 12, what is a² + b²? (a) 157 (b) 193 (c) 145 (d) 169
  34. 34.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)
  35. 35.What is the sum of the roots of x² − 14x + 48 = 0? (a) 48 (b) 13 (c) 15 (d) 14
  36. 36.If a + b = 13 and ab = 36, what is a² + b²? (a) 97 (b) 241 (c) 133 (d) 169
  37. 37.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6561 (b) 6571 (c) 6401 (d) 6241
  38. 38.What is the sum of the roots of x² − 11x + 24 = 0? (a) 24 (b) 11 (c) 12 (d) 10
  39. 39.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2126 (b) 2484 (c) 2916 (d) 2116
  40. 40.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 27) (b) (x + 1)(x + 11) (c) (x + 3)(x + 9) (d) (x − 3)(x − 9)

Answer key

  1. 9801
  2. 1
  3. 180
  4. 20
  5. 72
  6. 45
  7. 4096
  8. 1
  9. 56
  10. 13
  11. 122
  12. 1296
  13. 5 and 6
  14. 256
  15. 4
  16. 7
  17. (x + 7)(x + 9)
  18. 117
  19. 5
  20. 7 and 8
  21. (x + 4)(x + 8)
  22. (x + 7)(x + 8)
  23. 5329
  24. 17
  25. 11025
  26. 4
  27. 441
  28. (x + 8)(x + 9)
  29. 49
  30. 5 and 7
  31. (x + 4)(x + 7)
  32. 5476
  33. 145
  34. (x + 5)(x + 9)
  35. 14
  36. 97
  37. 6561
  38. 11
  39. 2116
  40. (x + 3)(x + 9)

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