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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 996 · 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 92²? (a) 8104 (b) 8474 (c) 8464 (d) 7744
  2. 2.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 925 (b) 625 (c) 1235 (d) 1225
  3. 3.If a + b = 9 and ab = 14, what is (a − b)²? (a) 53 (b) 81 (c) 5 (d) 25
  4. 4.What is the sum of the roots of x² − 8x + 15 = 0? (a) 8 (b) 7 (c) 9 (d) 15
  5. 5.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1531 (b) 1521 (c) 1681 (d) 1599
  6. 6.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x + 1)(x + 42) (c) (x + 6)(x + 7) (d) (x − 6)(x − 7)
  7. 7.Which is the correct factorisation of x² + 12x + 35? (a) (x − 5)(x − 7) (b) (x + 1)(x + 35) (c) (x + 1)(x + 11) (d) (x + 5)(x + 7)
  8. 8.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 441 (b) 361 (c) 451 (d) 401
  9. 9.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2714 (b) 2504 (c) 2304 (d) 2704
  10. 10.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 3616 (b) 4096 (c) 3136 (d) 4106
  11. 11.What are the roots of x² − 10x + 21 = 0? (a) 1 and 21 (b) 2 and 8 (c) 1 and 9 (d) 3 and 7
  12. 12.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 256 (b) 266 (c) 384 (d) 576
  13. 13.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 416 (b) 256 (c) 586 (d) 576
  14. 14.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 6409 (b) 5929 (c) 6889 (d) 6899
  15. 15.If a + b = 12 and ab = 32, what is (a − b)²? (a) 144 (b) 4 (c) 80 (d) 16
  16. 16.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3249 (b) 3969 (c) 3609 (d) 3979
  17. 17.If a + b = 10 and ab = 9, what is a² + b²? (a) 118 (b) 82 (c) 91 (d) 100
  18. 18.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1774 (b) 1444 (c) 1764 (d) 1604
  19. 19.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9035 (b) 9025 (c) 9975 (d) 11025
  20. 20.Which is the correct factorisation of x² + 9x + 20? (a) (x − 4)(x − 5) (b) (x + 1)(x + 8) (c) (x + 1)(x + 20) (d) (x + 4)(x + 5)
  21. 21.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 5635 (b) 4925 (c) 5625 (d) 4225
  22. 22.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 11025 (b) 10025 (c) 11035 (d) 9025
  23. 23.What are the roots of x² − 8x + 15 = 0? (a) 2 and 6 (b) 3 and 5 (c) 1 and 7 (d) 1 and 15
  24. 24.Which is the correct factorisation of x² + 8x + 12? (a) (x − 2)(x − 6) (b) (x + 1)(x + 7) (c) (x + 2)(x + 6) (d) (x + 1)(x + 12)
  25. 25.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 1166 (b) 916 (c) 676 (d) 1156
  26. 26.If a + b = 19 and ab = 84, what is a² + b²? (a) 193 (b) 361 (c) 277 (d) 529
  27. 27.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1859 (b) 1849 (c) 1609 (d) 1369
  28. 28.If a + b = 10 and ab = 21, what is (a − b)²? (a) 4 (b) 100 (c) 58 (d) 16
  29. 29.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6396 (b) 6724 (c) 6094 (d) 6084
  30. 30.If a + b = 13 and ab = 42, what is a² + b²? (a) 253 (b) 127 (c) 85 (d) 169
  31. 31.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 18) (b) (x − 3)(x − 6) (c) (x + 3)(x + 6) (d) (x + 1)(x + 8)
  32. 32.What are the roots of x² − 7x + 10 = 0? (a) 2 and 5 (b) 1 and 10 (c) 3 and 4 (d) 1 and 6
  33. 33.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 9226 (b) 9984 (c) 10816 (d) 9216
  34. 34.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 14 (c) 15 (d) 16
  35. 35.What is the product of the roots of x² − 13x + 36 = 0? (a) 35 (b) 37 (c) 13 (d) 36
  36. 36.What is the product of the roots of x² − 8x + 12 = 0? (a) 8 (b) 12 (c) 11 (d) 13
  37. 37.If a + b = 18 and ab = 77, what is a² + b²? (a) 478 (b) 247 (c) 170 (d) 324
  38. 38.If a + b = 16 and ab = 55, what is (a − b)²? (a) 6 (b) 36 (c) 256 (d) 146
  39. 39.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3969 (b) 3591 (c) 3259 (d) 3249
  40. 40.If a + b = 13 and ab = 30, what is a² + b²? (a) 139 (b) 109 (c) 169 (d) 229

Answer key

  1. 8464
  2. 1225
  3. 25
  4. 8
  5. 1521
  6. (x + 6)(x + 7)
  7. (x + 5)(x + 7)
  8. 441
  9. 2704
  10. 4096
  11. 3 and 7
  12. 256
  13. 576
  14. 6889
  15. 16
  16. 3969
  17. 82
  18. 1764
  19. 9025
  20. (x + 4)(x + 5)
  21. 5625
  22. 11025
  23. 3 and 5
  24. (x + 2)(x + 6)
  25. 1156
  26. 193
  27. 1849
  28. 16
  29. 6084
  30. 85
  31. (x + 3)(x + 6)
  32. 2 and 5
  33. 9216
  34. 15
  35. 36
  36. 12
  37. 170
  38. 36
  39. 3249
  40. 109

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