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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 196 · 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.What is the sum of the roots of x² − 12x + 35 = 0? (a) 35 (b) 13 (c) 11 (d) 12
  2. 2.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1936 (b) 1584 (c) 1306 (d) 1296
  3. 3.If a + b = 17 and ab = 72, what is a² + b²? (a) 145 (b) 289 (c) 217 (d) 433
  4. 4.If a + b = 17 and ab = 66, what is a² + b²? (a) 289 (b) 421 (c) 223 (d) 157
  5. 5.What is the sum of the roots of x² − 13x + 40 = 0? (a) 14 (b) 13 (c) 40 (d) 12
  6. 6.If a + b = 8 and ab = 15, what is a² + b²? (a) 34 (b) 94 (c) 64 (d) 49
  7. 7.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 2475 (b) 3025 (c) 2035 (d) 2025
  8. 8.What is the sum of the roots of x² − 12x + 27 = 0? (a) 12 (b) 11 (c) 27 (d) 13
  9. 9.If a + b = 20 and ab = 96, what is a² + b²? (a) 400 (b) 592 (c) 304 (d) 208
  10. 10.If a + b = 20 and ab = 96, what is (a − b)²? (a) 16 (b) 400 (c) 4 (d) 208
  11. 11.What are the roots of x² − 15x + 56 = 0? (a) 2 and 28 (b) 7 and 8 (c) 1 and 56 (d) 1 and 14
  12. 12.What is the sum of the roots of x² − 16x + 63 = 0? (a) 17 (b) 16 (c) 15 (d) 63
  13. 13.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 8096 (b) 7754 (c) 7744 (d) 8464
  14. 14.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1444 (b) 1596 (c) 1454 (d) 1764
  15. 15.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2209 (b) 2809 (c) 2509 (d) 2819
  16. 16.What is the sum of the roots of x² − 11x + 28 = 0? (a) 12 (b) 10 (c) 11 (d) 28
  17. 17.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2714 (b) 2304 (c) 2704 (d) 2504
  18. 18.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3591 (b) 3259 (c) 3249 (d) 3969
  19. 19.Which is the correct factorisation of x² + 9x + 14? (a) (x − 2)(x − 7) (b) (x + 1)(x + 14) (c) (x + 1)(x + 8) (d) (x + 2)(x + 7)
  20. 20.What are the roots of x² − 9x + 18 = 0? (a) 1 and 18 (b) 3 and 6 (c) 1 and 8 (d) 2 and 9
  21. 21.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 12) (b) (x + 5)(x + 8) (c) (x − 5)(x − 8) (d) (x + 1)(x + 40)
  22. 22.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 586 (b) 576 (c) 256 (d) 416
  23. 23.What are the roots of x² − 11x + 24 = 0? (a) 1 and 24 (b) 1 and 10 (c) 3 and 8 (d) 2 and 12
  24. 24.What is the sum of the roots of x² − 14x + 48 = 0? (a) 13 (b) 48 (c) 15 (d) 14
  25. 25.If a + b = 6 and ab = 8, what is a² + b²? (a) 20 (b) 52 (c) 36 (d) 28
  26. 26.If a + b = 17 and ab = 60, what is a² + b²? (a) 409 (b) 169 (c) 289 (d) 229
  27. 27.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x + 1)(x + 48) (c) (x + 6)(x + 8) (d) (x − 6)(x − 8)
  28. 28.If a + b = 18 and ab = 77, what is a² + b²? (a) 478 (b) 324 (c) 170 (d) 247
  29. 29.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 9604 (b) 10414 (c) 10004 (d) 10404
  30. 30.If a + b = 14 and ab = 45, what is a² + b²? (a) 151 (b) 106 (c) 286 (d) 196
  31. 31.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1774 (b) 1764 (c) 1604 (d) 1444
  32. 32.If a + b = 18 and ab = 72, what is a² + b²? (a) 252 (b) 468 (c) 324 (d) 180
  33. 33.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 8075 (b) 7235 (c) 7225 (d) 9025
  34. 34.If a + b = 16 and ab = 60, what is (a − b)²? (a) 136 (b) 256 (c) 4 (d) 16
  35. 35.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 1089 (b) 891 (c) 729 (d) 739
  36. 36.If a + b = 19 and ab = 84, what is (a − b)²? (a) 361 (b) 25 (c) 5 (d) 193
  37. 37.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 71 (c) 17 (d) 72
  38. 38.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 72) (c) (x + 1)(x + 16) (d) (x + 8)(x + 9)
  39. 39.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 539 (b) 529 (c) 409 (d) 289
  40. 40.What is the sum of the roots of x² − 13x + 42 = 0? (a) 13 (b) 12 (c) 42 (d) 14

Answer key

  1. 12
  2. 1296
  3. 145
  4. 157
  5. 13
  6. 34
  7. 2025
  8. 12
  9. 208
  10. 16
  11. 7 and 8
  12. 16
  13. 7744
  14. 1444
  15. 2809
  16. 11
  17. 2704
  18. 3249
  19. (x + 2)(x + 7)
  20. 3 and 6
  21. (x + 5)(x + 8)
  22. 576
  23. 3 and 8
  24. 14
  25. 20
  26. 169
  27. (x + 6)(x + 8)
  28. 170
  29. 10404
  30. 106
  31. 1764
  32. 180
  33. 7225
  34. 16
  35. 729
  36. 25
  37. 72
  38. (x + 8)(x + 9)
  39. 529
  40. 13

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