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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 108 · 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 17²? (a) 299 (b) 289 (c) 529 (d) 391
  2. 2.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8101 (b) 7921 (c) 8291 (d) 8281
  3. 3.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2484 (b) 2916 (c) 2126 (d) 2116
  4. 4.If a + b = 13 and ab = 22, what is (a − b)²? (a) 169 (b) 125 (c) 81 (d) 9
  5. 5.What is the product of the roots of x² − 15x + 56 = 0? (a) 55 (b) 15 (c) 57 (d) 56
  6. 6.If a + b = 14 and ab = 48, what is (a − b)²? (a) 4 (b) 2 (c) 100 (d) 196
  7. 7.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 2025 (b) 2475 (c) 2035 (d) 3025
  8. 8.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 12) (b) (x + 1)(x + 6) (c) (x + 3)(x + 4) (d) (x − 3)(x − 4)
  9. 9.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 10826 (b) 10016 (c) 9216 (d) 10816
  10. 10.What are the roots of x² − 15x + 56 = 0? (a) 1 and 14 (b) 1 and 56 (c) 2 and 28 (d) 7 and 8
  11. 11.What is the sum of the roots of x² − 10x + 16 = 0? (a) 10 (b) 9 (c) 16 (d) 11
  12. 12.Which is the correct factorisation of x² + 12x + 32? (a) (x + 4)(x + 8) (b) (x + 1)(x + 32) (c) (x − 4)(x − 8) (d) (x + 1)(x + 11)
  13. 13.Which is the correct factorisation of x² + 8x + 15? (a) (x − 3)(x − 5) (b) (x + 1)(x + 15) (c) (x + 1)(x + 7) (d) (x + 3)(x + 5)
  14. 14.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 10) (b) (x − 2)(x − 5) (c) (x + 1)(x + 6) (d) (x + 2)(x + 5)
  15. 15.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2499 (b) 2601 (c) 2411 (d) 2401
  16. 16.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 5476 (b) 4884 (c) 4356 (d) 4366
  17. 17.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 686 (b) 884 (c) 676 (d) 1156
  18. 18.What is the product of the roots of x² − 17x + 72 = 0? (a) 72 (b) 71 (c) 17 (d) 73
  19. 19.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2496 (b) 2704 (c) 2304 (d) 2314
  20. 20.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 10) (b) (x + 1)(x + 24) (c) (x − 3)(x − 8) (d) (x + 3)(x + 8)
  21. 21.If a + b = 3 and ab = 2, what is a² + b²? (a) 13 (b) 5 (c) 7 (d) 9
  22. 22.What are the roots of x² − 5x + 6 = 0? (a) 1 and 4 (b) 1 and 6 (c) 2 and 3 (d) 2 and 4
  23. 23.If a + b = 17 and ab = 66, what is a² + b²? (a) 421 (b) 157 (c) 223 (d) 289
  24. 24.If a + b = 13 and ab = 40, what is a² + b²? (a) 249 (b) 129 (c) 89 (d) 169
  25. 25.If a + b = 19 and ab = 90, what is (a − b)²? (a) 181 (b) 1 (c) 271 (d) 361
  26. 26.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 10816 (b) 9226 (c) 9984 (d) 9216
  27. 27.If a + b = 8 and ab = 15, what is a² + b²? (a) 49 (b) 34 (c) 94 (d) 64
  28. 28.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 16) (c) (x + 8)(x + 9) (d) (x + 1)(x + 72)
  29. 29.What is the product of the roots of x² − 7x + 10 = 0? (a) 10 (b) 7 (c) 11 (d) 9
  30. 30.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)
  31. 31.If a + b = 11 and ab = 30, what is (a − b)²? (a) 121 (b) 91 (c) 1 (d) 61
  32. 32.If a + b = 6 and ab = 8, what is a² + b²? (a) 52 (b) 36 (c) 20 (d) 28
  33. 33.If a + b = 17 and ab = 72, what is (a − b)²? (a) 289 (b) 217 (c) 145 (d) 1
  34. 34.If a + b = 18 and ab = 80, what is a² + b²? (a) 484 (b) 244 (c) 324 (d) 164
  35. 35.What is the product of the roots of x² − 7x + 12 = 0? (a) 11 (b) 7 (c) 12 (d) 13
  36. 36.If a + b = 10 and ab = 21, what is a² + b²? (a) 58 (b) 79 (c) 142 (d) 100
  37. 37.What are the roots of x² − 16x + 63 = 0? (a) 1 and 63 (b) 7 and 9 (c) 3 and 21 (d) 1 and 15
  38. 38.If a + b = 13 and ab = 42, what is (a − b)²? (a) 1 (b) 169 (c) 127 (d) 85
  39. 39.What is the sum of the roots of x² − 10x + 24 = 0? (a) 24 (b) 11 (c) 9 (d) 10
  40. 40.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 16) (b) (x + 2)(x + 8) (c) (x − 2)(x − 8) (d) (x + 1)(x + 9)

Answer key

  1. 289
  2. 8281
  3. 2116
  4. 81
  5. 56
  6. 4
  7. 2025
  8. (x + 3)(x + 4)
  9. 10816
  10. 7 and 8
  11. 10
  12. (x + 4)(x + 8)
  13. (x + 3)(x + 5)
  14. (x + 2)(x + 5)
  15. 2401
  16. 4356
  17. 676
  18. 72
  19. 2304
  20. (x + 3)(x + 8)
  21. 5
  22. 2 and 3
  23. 157
  24. 89
  25. 1
  26. 9216
  27. 34
  28. (x + 8)(x + 9)
  29. 10
  30. (x + 7)(x + 9)
  31. 1
  32. 20
  33. 1
  34. 164
  35. 12
  36. 58
  37. 7 and 9
  38. 1
  39. 10
  40. (x + 2)(x + 8)

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