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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 11 · 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 26²? (a) 686 (b) 676 (c) 884 (d) 1156
  2. 2.If a + b = 8 and ab = 15, what is (a − b)²? (a) 64 (b) 2 (c) 34 (d) 4
  3. 3.If a + b = 15 and ab = 56, what is a² + b²? (a) 225 (b) 337 (c) 113 (d) 169
  4. 4.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 529 (b) 299 (c) 289 (d) 391
  5. 5.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2496 (b) 2304 (c) 2704 (d) 2314
  6. 6.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 18) (b) (x − 2)(x − 9) (c) (x + 1)(x + 10) (d) (x + 2)(x + 9)
  7. 7.What are the roots of x² − 16x + 63 = 0? (a) 1 and 15 (b) 1 and 63 (c) 3 and 21 (d) 7 and 9
  8. 8.If a + b = 17 and ab = 70, what is a² + b²? (a) 149 (b) 219 (c) 289 (d) 429
  9. 9.What is the product of the roots of x² − 7x + 10 = 0? (a) 11 (b) 9 (c) 7 (d) 10
  10. 10.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 961 (b) 841 (c) 901 (d) 971
  11. 11.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 1625 (b) 1225 (c) 2025 (d) 2035
  12. 12.If a + b = 5 and ab = 4, what is a² + b²? (a) 17 (b) 25 (c) 33 (d) 21
  13. 13.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1296 (b) 1936 (c) 1946 (d) 1616
  14. 14.Which is the correct factorisation of x² + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 5)(x + 6) (c) (x + 1)(x + 10) (d) (x + 1)(x + 30)
  15. 15.If a + b = 19 and ab = 88, what is a² + b²? (a) 361 (b) 185 (c) 537 (d) 273
  16. 16.What is the sum of the roots of x² − 12x + 32 = 0? (a) 32 (b) 11 (c) 13 (d) 12
  17. 17.If a + b = 6 and ab = 8, what is a² + b²? (a) 36 (b) 20 (c) 52 (d) 28
  18. 18.What is the sum of the roots of x² − 9x + 14 = 0? (a) 10 (b) 8 (c) 14 (d) 9
  19. 19.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x − 7)(x − 9) (c) (x + 1)(x + 15) (d) (x + 7)(x + 9)
  20. 20.If a + b = 19 and ab = 90, what is (a − b)²? (a) 361 (b) 271 (c) 181 (d) 1
  21. 21.What is the sum of the roots of x² − 10x + 21 = 0? (a) 9 (b) 21 (c) 10 (d) 11
  22. 22.If a + b = 8 and ab = 12, what is (a − b)²? (a) 40 (b) 16 (c) 64 (d) 4
  23. 23.If a + b = 18 and ab = 80, what is a² + b²? (a) 324 (b) 484 (c) 244 (d) 164
  24. 24.If a + b = 9 and ab = 8, what is a² + b²? (a) 81 (b) 97 (c) 65 (d) 73
  25. 25.If a + b = 9 and ab = 20, what is (a − b)²? (a) 61 (b) 1 (c) 81 (d) 41
  26. 26.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3481 (b) 3601 (c) 3731 (d) 3721
  27. 27.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 5194 (b) 5184 (c) 4904 (d) 4624
  28. 28.What is the product of the roots of x² − 10x + 21 = 0? (a) 20 (b) 10 (c) 21 (d) 22
  29. 29.If a + b = 13 and ab = 12, what is a² + b²? (a) 169 (b) 157 (c) 193 (d) 145
  30. 30.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 676 (b) 1156 (c) 1166 (d) 916
  31. 31.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 494 (b) 324 (c) 404 (d) 484
  32. 32.If a + b = 11 and ab = 10, what is a² + b²? (a) 111 (b) 101 (c) 141 (d) 121
  33. 33.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 8084 (b) 7406 (c) 8836 (d) 7396
  34. 34.If a + b = 12 and ab = 27, what is a² + b²? (a) 117 (b) 144 (c) 90 (d) 198
  35. 35.If a + b = 20 and ab = 96, what is a² + b²? (a) 400 (b) 592 (c) 304 (d) 208
  36. 36.If a + b = 8 and ab = 12, what is a² + b²? (a) 40 (b) 88 (c) 52 (d) 64
  37. 37.If a + b = 16 and ab = 60, what is (a − b)²? (a) 256 (b) 4 (c) 16 (d) 136
  38. 38.Which is the correct factorisation of x² + 13x + 40? (a) (x + 5)(x + 8) (b) (x − 5)(x − 8) (c) (x + 1)(x + 40) (d) (x + 1)(x + 12)
  39. 39.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1681 (b) 1599 (c) 1531 (d) 1521
  40. 40.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 17 (c) 72 (d) 71

Answer key

  1. 676
  2. 4
  3. 113
  4. 289
  5. 2304
  6. (x + 2)(x + 9)
  7. 7 and 9
  8. 149
  9. 10
  10. 961
  11. 2025
  12. 17
  13. 1936
  14. (x + 5)(x + 6)
  15. 185
  16. 12
  17. 20
  18. 9
  19. (x + 7)(x + 9)
  20. 1
  21. 10
  22. 16
  23. 164
  24. 65
  25. 1
  26. 3721
  27. 5184
  28. 21
  29. 145
  30. 1156
  31. 484
  32. 101
  33. 7396
  34. 90
  35. 208
  36. 40
  37. 16
  38. (x + 5)(x + 8)
  39. 1521
  40. 72

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