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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 618 · 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 31²? (a) 971 (b) 901 (c) 841 (d) 961
  2. 2.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1681 (b) 1601 (c) 1691 (d) 1521
  3. 3.If a + b = 19 and ab = 84, what is (a − b)²? (a) 5 (b) 193 (c) 361 (d) 25
  4. 4.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 676 (b) 686 (c) 884 (d) 1156
  5. 5.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1936 (b) 1306 (c) 1296 (d) 1584
  6. 6.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x + 6)(x + 8) (c) (x − 6)(x − 8) (d) (x + 1)(x + 48)
  7. 7.Which is the correct factorisation of x² + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 1)(x + 6) (c) (x + 3)(x + 4) (d) (x + 1)(x + 12)
  8. 8.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2714 (b) 2304 (c) 2504 (d) 2704
  9. 9.If a + b = 20 and ab = 96, what is a² + b²? (a) 400 (b) 304 (c) 592 (d) 208
  10. 10.If a + b = 17 and ab = 72, what is a² + b²? (a) 433 (b) 289 (c) 217 (d) 145
  11. 11.Which is the correct factorisation of x² + 11x + 24? (a) (x + 3)(x + 8) (b) (x + 1)(x + 24) (c) (x − 3)(x − 8) (d) (x + 1)(x + 10)
  12. 12.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 635 (b) 625 (c) 425 (d) 225
  13. 13.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3721 (b) 3599 (c) 3491 (d) 3481
  14. 14.If a + b = 20 and ab = 99, what is a² + b²? (a) 400 (b) 202 (c) 598 (d) 301
  15. 15.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 529 (b) 299 (c) 391 (d) 289
  16. 16.If a + b = 15 and ab = 44, what is (a − b)²? (a) 49 (b) 7 (c) 137 (d) 225
  17. 17.If a + b = 6 and ab = 8, what is (a − b)²? (a) 4 (b) 2 (c) 36 (d) 20
  18. 18.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 404 (b) 324 (c) 484 (d) 494
  19. 19.If a + b = 9 and ab = 8, what is a² + b²? (a) 81 (b) 97 (c) 73 (d) 65
  20. 20.What is the sum of the roots of x² − 11x + 30 = 0? (a) 12 (b) 30 (c) 10 (d) 11
  21. 21.Which is the correct factorisation of x² + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 1)(x + 30) (c) (x + 1)(x + 10) (d) (x + 5)(x + 6)
  22. 22.If a + b = 19 and ab = 88, what is a² + b²? (a) 273 (b) 537 (c) 185 (d) 361
  23. 23.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)
  24. 24.If a + b = 16 and ab = 63, what is a² + b²? (a) 130 (b) 256 (c) 193 (d) 382
  25. 25.If a + b = 15 and ab = 56, what is a² + b²? (a) 225 (b) 337 (c) 113 (d) 169
  26. 26.If a + b = 8 and ab = 12, what is (a − b)²? (a) 64 (b) 40 (c) 16 (d) 4
  27. 27.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 7921 (b) 8291 (c) 8101 (d) 8281
  28. 28.If a + b = 5 and ab = 4, what is a² + b²? (a) 17 (b) 25 (c) 33 (d) 21
  29. 29.Which is the correct factorisation of x² + 8x + 15? (a) (x − 3)(x − 5) (b) (x + 1)(x + 15) (c) (x + 3)(x + 5) (d) (x + 1)(x + 7)
  30. 30.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 18) (b) (x − 3)(x − 6) (c) (x + 1)(x + 8) (d) (x + 3)(x + 6)
  31. 31.What are the roots of x² − 9x + 20 = 0? (a) 1 and 8 (b) 2 and 10 (c) 1 and 20 (d) 4 and 5
  32. 32.Which is the correct factorisation of x² + 13x + 42? (a) (x + 6)(x + 7) (b) (x − 6)(x − 7) (c) (x + 1)(x + 12) (d) (x + 1)(x + 42)
  33. 33.If a + b = 15 and ab = 50, what is (a − b)²? (a) 125 (b) 5 (c) 25 (d) 225
  34. 34.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 399 (b) 441 (c) 371 (d) 361
  35. 35.If a + b = 7 and ab = 12, what is a² + b²? (a) 73 (b) 25 (c) 49 (d) 37
  36. 36.Which is the correct factorisation of x² + 11x + 18? (a) (x − 2)(x − 9) (b) (x + 1)(x + 18) (c) (x + 1)(x + 10) (d) (x + 2)(x + 9)
  37. 37.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x + 1)(x + 56) (c) (x − 7)(x − 8) (d) (x + 1)(x + 14)
  38. 38.If a + b = 13 and ab = 40, what is a² + b²? (a) 129 (b) 89 (c) 169 (d) 249
  39. 39.If a + b = 16 and ab = 48, what is a² + b²? (a) 256 (b) 352 (c) 160 (d) 208
  40. 40.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 256 (b) 416 (c) 576 (d) 586

Answer key

  1. 961
  2. 1681
  3. 25
  4. 676
  5. 1296
  6. (x + 6)(x + 8)
  7. (x + 3)(x + 4)
  8. 2704
  9. 208
  10. 145
  11. (x + 3)(x + 8)
  12. 625
  13. 3481
  14. 202
  15. 289
  16. 49
  17. 4
  18. 484
  19. 65
  20. 11
  21. (x + 5)(x + 6)
  22. 185
  23. (x + 8)(x + 9)
  24. 130
  25. 113
  26. 16
  27. 8281
  28. 17
  29. (x + 3)(x + 5)
  30. (x + 3)(x + 6)
  31. 4 and 5
  32. (x + 6)(x + 7)
  33. 25
  34. 361
  35. 25
  36. (x + 2)(x + 9)
  37. (x + 7)(x + 8)
  38. 89
  39. 160
  40. 576

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