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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 161 · 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 = 8 and ab = 15, what is a² + b²? (a) 34 (b) 49 (c) 64 (d) 94
  2. 2.What is the sum of the roots of x² − 15x + 56 = 0? (a) 14 (b) 16 (c) 56 (d) 15
  3. 3.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3601 (b) 3721 (c) 3731 (d) 3481
  4. 4.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2809 (b) 2491 (c) 2219 (d) 2209
  5. 5.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 1575 (b) 2025 (c) 1235 (d) 1225
  6. 6.What is the sum of the roots of x² − 17x + 72 = 0? (a) 17 (b) 18 (c) 16 (d) 72
  7. 7.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x − 6)(x − 8) (c) (x + 6)(x + 8) (d) (x + 1)(x + 13)
  8. 8.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 484 (b) 404 (c) 324 (d) 494
  9. 9.What are the roots of x² − 15x + 56 = 0? (a) 2 and 28 (b) 7 and 8 (c) 1 and 14 (d) 1 and 56
  10. 10.If a + b = 21 and ab = 108, what is (a − b)²? (a) 3 (b) 225 (c) 9 (d) 441
  11. 11.If a + b = 15 and ab = 56, what is (a − b)²? (a) 169 (b) 225 (c) 113 (d) 1
  12. 12.What are the roots of x² − 17x + 72 = 0? (a) 2 and 36 (b) 1 and 72 (c) 8 and 9 (d) 1 and 16
  13. 13.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2704 (b) 2504 (c) 2304 (d) 2714
  14. 14.What are the roots of x² − 9x + 20 = 0? (a) 4 and 5 (b) 2 and 10 (c) 1 and 8 (d) 1 and 20
  15. 15.What is the sum of the roots of x² − 13x + 36 = 0? (a) 13 (b) 36 (c) 12 (d) 14
  16. 16.What are the roots of x² − 10x + 21 = 0? (a) 1 and 9 (b) 3 and 7 (c) 2 and 8 (d) 1 and 21
  17. 17.If a + b = 20 and ab = 96, what is (a − b)²? (a) 16 (b) 400 (c) 208 (d) 4
  18. 18.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 7) (b) (x + 1)(x + 15) (c) (x + 3)(x + 5) (d) (x − 3)(x − 5)
  19. 19.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 40) (b) (x − 5)(x − 8) (c) (x + 5)(x + 8) (d) (x + 1)(x + 12)
  20. 20.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 72) (c) (x + 8)(x + 9) (d) (x + 1)(x + 16)
  21. 21.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 7406 (b) 8836 (c) 7396 (d) 8084
  22. 22.If a + b = 9 and ab = 8, what is a² + b²? (a) 73 (b) 81 (c) 65 (d) 97
  23. 23.If a + b = 10 and ab = 21, what is a² + b²? (a) 142 (b) 100 (c) 58 (d) 79
  24. 24.What is the product of the roots of x² − 15x + 56 = 0? (a) 55 (b) 15 (c) 56 (d) 57
  25. 25.If a + b = 10 and ab = 16, what is a² + b²? (a) 68 (b) 84 (c) 132 (d) 100
  26. 26.What is the product of the roots of x² − 13x + 40 = 0? (a) 13 (b) 41 (c) 39 (d) 40
  27. 27.If a + b = 10 and ab = 9, what is (a − b)²? (a) 8 (b) 82 (c) 64 (d) 100
  28. 28.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2601 (b) 2499 (c) 2411 (d) 2401
  29. 29.What is the product of the roots of x² − 11x + 18 = 0? (a) 18 (b) 19 (c) 11 (d) 17
  30. 30.If a + b = 19 and ab = 84, what is a² + b²? (a) 529 (b) 361 (c) 277 (d) 193
  31. 31.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 2035 (b) 1225 (c) 2025 (d) 1625
  32. 32.What is the product of the roots of x² − 8x + 15 = 0? (a) 15 (b) 8 (c) 16 (d) 14
  33. 33.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 384 (b) 576 (c) 256 (d) 266
  34. 34.If a + b = 14 and ab = 24, what is a² + b²? (a) 148 (b) 172 (c) 196 (d) 244
  35. 35.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 8) (b) (x + 2)(x + 4) (c) (x − 2)(x − 4) (d) (x + 1)(x + 5)
  36. 36.What is the sum of the roots of x² − 10x + 24 = 0? (a) 9 (b) 11 (c) 10 (d) 24
  37. 37.If a + b = 12 and ab = 11, what is a² + b²? (a) 166 (b) 144 (c) 133 (d) 122
  38. 38.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 901 (b) 841 (c) 961 (d) 971
  39. 39.If a + b = 13 and ab = 12, what is a² + b²? (a) 145 (b) 169 (c) 193 (d) 157
  40. 40.Which is the correct factorisation of x² + 9x + 18? (a) (x − 3)(x − 6) (b) (x + 3)(x + 6) (c) (x + 1)(x + 8) (d) (x + 1)(x + 18)

Answer key

  1. 34
  2. 15
  3. 3721
  4. 2209
  5. 1225
  6. 17
  7. (x + 6)(x + 8)
  8. 484
  9. 7 and 8
  10. 9
  11. 1
  12. 8 and 9
  13. 2704
  14. 4 and 5
  15. 13
  16. 3 and 7
  17. 16
  18. (x + 3)(x + 5)
  19. (x + 5)(x + 8)
  20. (x + 8)(x + 9)
  21. 7396
  22. 65
  23. 58
  24. 56
  25. 68
  26. 40
  27. 64
  28. 2401
  29. 18
  30. 193
  31. 2025
  32. 15
  33. 256
  34. 148
  35. (x + 2)(x + 4)
  36. 10
  37. 122
  38. 961
  39. 145
  40. (x + 3)(x + 6)

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