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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 320 · 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 65²? (a) 4875 (b) 4235 (c) 4225 (d) 5625
  2. 2.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)
  3. 3.What is the product of the roots of x² − 8x + 12 = 0? (a) 8 (b) 13 (c) 12 (d) 11
  4. 4.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 3136 (b) 3584 (c) 3146 (d) 4096
  5. 5.If a + b = 15 and ab = 54, what is a² + b²? (a) 333 (b) 117 (c) 225 (d) 171
  6. 6.If a + b = 12 and ab = 35, what is a² + b²? (a) 109 (b) 214 (c) 74 (d) 144
  7. 7.If a + b = 17 and ab = 70, what is (a − b)²? (a) 3 (b) 289 (c) 149 (d) 9
  8. 8.If a + b = 17 and ab = 72, what is a² + b²? (a) 289 (b) 145 (c) 217 (d) 433
  9. 9.If a + b = 19 and ab = 84, what is a² + b²? (a) 529 (b) 277 (c) 193 (d) 361
  10. 10.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x + 1)(x + 15) (c) (x − 7)(x − 9) (d) (x + 7)(x + 9)
  11. 11.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 625 (b) 1235 (c) 925 (d) 1225
  12. 12.If a + b = 9 and ab = 8, what is a² + b²? (a) 65 (b) 81 (c) 97 (d) 73
  13. 13.What is the product of the roots of x² − 12x + 32 = 0? (a) 32 (b) 33 (c) 12 (d) 31
  14. 14.What is the sum of the roots of x² − 10x + 16 = 0? (a) 16 (b) 9 (c) 11 (d) 10
  15. 15.If a + b = 3 and ab = 2, what is a² + b²? (a) 13 (b) 7 (c) 5 (d) 9
  16. 16.If a + b = 14 and ab = 45, what is a² + b²? (a) 286 (b) 196 (c) 151 (d) 106
  17. 17.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 14) (c) (x + 1)(x + 56) (d) (x + 7)(x + 8)
  18. 18.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 18 (c) 17 (d) 72
  19. 19.If a + b = 20 and ab = 99, what is a² + b²? (a) 598 (b) 400 (c) 301 (d) 202
  20. 20.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 371 (b) 399 (c) 441 (d) 361
  21. 21.If a + b = 10 and ab = 9, what is (a − b)²? (a) 82 (b) 64 (c) 8 (d) 100
  22. 22.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 5486 (b) 4356 (c) 4916 (d) 5476
  23. 23.If a + b = 19 and ab = 88, what is a² + b²? (a) 361 (b) 537 (c) 185 (d) 273
  24. 24.What is the product of the roots of x² − 17x + 72 = 0? (a) 72 (b) 73 (c) 71 (d) 17
  25. 25.What is the sum of the roots of x² − 16x + 63 = 0? (a) 15 (b) 63 (c) 16 (d) 17
  26. 26.What is the sum of the roots of x² − 15x + 56 = 0? (a) 14 (b) 15 (c) 16 (d) 56
  27. 27.If a + b = 17 and ab = 60, what is (a − b)²? (a) 7 (b) 49 (c) 289 (d) 169
  28. 28.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 15) (b) (x + 3)(x + 5) (c) (x + 1)(x + 7) (d) (x − 3)(x − 5)
  29. 29.If a + b = 18 and ab = 77, what is a² + b²? (a) 478 (b) 324 (c) 170 (d) 247
  30. 30.If a + b = 13 and ab = 42, what is a² + b²? (a) 127 (b) 169 (c) 253 (d) 85
  31. 31.If a + b = 12 and ab = 27, what is a² + b²? (a) 144 (b) 90 (c) 117 (d) 198
  32. 32.Which is the correct factorisation of x² + 9x + 20? (a) (x + 4)(x + 5) (b) (x + 1)(x + 20) (c) (x − 4)(x − 5) (d) (x + 1)(x + 8)
  33. 33.If a + b = 11 and ab = 28, what is (a − b)²? (a) 121 (b) 3 (c) 9 (d) 65
  34. 34.Which is the correct factorisation of x² + 9x + 14? (a) (x + 2)(x + 7) (b) (x − 2)(x − 7) (c) (x + 1)(x + 8) (d) (x + 1)(x + 14)
  35. 35.Which is the correct factorisation of x² + 11x + 24? (a) (x − 3)(x − 8) (b) (x + 1)(x + 10) (c) (x + 1)(x + 24) (d) (x + 3)(x + 8)
  36. 36.What are the roots of x² − 10x + 16 = 0? (a) 3 and 7 (b) 1 and 16 (c) 1 and 9 (d) 2 and 8
  37. 37.What are the roots of x² − 12x + 35 = 0? (a) 1 and 35 (b) 1 and 11 (c) 5 and 7 (d) 2 and 10
  38. 38.If a + b = 10 and ab = 24, what is a² + b²? (a) 52 (b) 100 (c) 76 (d) 148
  39. 39.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 9) (b) (x + 1)(x + 21) (c) (x − 3)(x − 7) (d) (x + 3)(x + 7)
  40. 40.What is the sum of the roots of x² − 8x + 15 = 0? (a) 9 (b) 15 (c) 8 (d) 7

Answer key

  1. 4225
  2. (x + 4)(x + 6)
  3. 12
  4. 3136
  5. 117
  6. 74
  7. 9
  8. 145
  9. 193
  10. (x + 7)(x + 9)
  11. 1225
  12. 65
  13. 32
  14. 10
  15. 5
  16. 106
  17. (x + 7)(x + 8)
  18. 17
  19. 202
  20. 361
  21. 64
  22. 5476
  23. 185
  24. 72
  25. 16
  26. 15
  27. 49
  28. (x + 3)(x + 5)
  29. 170
  30. 85
  31. 90
  32. (x + 4)(x + 5)
  33. 9
  34. (x + 2)(x + 7)
  35. (x + 3)(x + 8)
  36. 2 and 8
  37. 5 and 7
  38. 52
  39. (x + 3)(x + 7)
  40. 8

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