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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 326 · 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 89²? (a) 7931 (b) 8099 (c) 7921 (d) 8281
  2. 2.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 494 (b) 404 (c) 484 (d) 324
  3. 3.What is the product of the roots of x² − 16x + 63 = 0? (a) 62 (b) 16 (c) 63 (d) 64
  4. 4.What is the product of the roots of x² − 11x + 30 = 0? (a) 11 (b) 30 (c) 31 (d) 29
  5. 5.If a + b = 10 and ab = 21, what is a² + b²? (a) 100 (b) 79 (c) 58 (d) 142
  6. 6.If a + b = 9 and ab = 14, what is (a − b)²? (a) 25 (b) 81 (c) 53 (d) 5
  7. 7.Which is the correct factorisation of x² + 12x + 35? (a) (x + 5)(x + 7) (b) (x + 1)(x + 11) (c) (x − 5)(x − 7) (d) (x + 1)(x + 35)
  8. 8.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)
  9. 9.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 289 (b) 539 (c) 529 (d) 409
  10. 10.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 1)(x + 16) (c) (x + 8)(x + 9) (d) (x − 8)(x − 9)
  11. 11.If a + b = 13 and ab = 30, what is a² + b²? (a) 169 (b) 229 (c) 139 (d) 109
  12. 12.Which is the correct factorisation of x² + 5x + 6? (a) (x − 2)(x − 3) (b) (x + 1)(x + 4) (c) (x + 2)(x + 3) (d) (x + 1)(x + 6)
  13. 13.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)
  14. 14.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 12) (b) (x − 4)(x − 9) (c) (x + 4)(x + 9) (d) (x + 1)(x + 36)
  15. 15.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 7921 (b) 8291 (c) 8281 (d) 8101
  16. 16.What are the roots of x² − 7x + 12 = 0? (a) 1 and 12 (b) 2 and 6 (c) 3 and 4 (d) 1 and 6
  17. 17.If a + b = 16 and ab = 60, what is (a − b)²? (a) 136 (b) 256 (c) 4 (d) 16
  18. 18.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)
  19. 19.If a + b = 9 and ab = 20, what is (a − b)²? (a) 81 (b) 61 (c) 41 (d) 1
  20. 20.What is the product of the roots of x² − 12x + 35 = 0? (a) 12 (b) 34 (c) 36 (d) 35
  21. 21.If a + b = 17 and ab = 72, what is a² + b²? (a) 217 (b) 433 (c) 145 (d) 289
  22. 22.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2516 (b) 2926 (c) 2116 (d) 2916
  23. 23.If a + b = 19 and ab = 88, what is a² + b²? (a) 185 (b) 537 (c) 361 (d) 273
  24. 24.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 416 (b) 256 (c) 586 (d) 576
  25. 25.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 7225 (b) 8075 (c) 9025 (d) 7235
  26. 26.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 8084 (b) 8836 (c) 7406 (d) 7396
  27. 27.What is the product of the roots of x² − 12x + 27 = 0? (a) 26 (b) 28 (c) 27 (d) 12
  28. 28.What is the sum of the roots of x² − 11x + 30 = 0? (a) 12 (b) 10 (c) 11 (d) 30
  29. 29.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3481 (b) 3721 (c) 3491 (d) 3599
  30. 30.What is the product of the roots of x² − 15x + 56 = 0? (a) 57 (b) 15 (c) 56 (d) 55
  31. 31.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 7) (b) (x + 3)(x + 5) (c) (x + 1)(x + 15) (d) (x − 3)(x − 5)
  32. 32.What is the product of the roots of x² − 14x + 48 = 0? (a) 49 (b) 14 (c) 47 (d) 48
  33. 33.If a + b = 17 and ab = 72, what is (a − b)²? (a) 145 (b) 1 (c) 217 (d) 289
  34. 34.If a + b = 14 and ab = 45, what is a² + b²? (a) 196 (b) 286 (c) 151 (d) 106
  35. 35.If a + b = 16 and ab = 48, what is (a − b)²? (a) 64 (b) 8 (c) 160 (d) 256
  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.If a + b = 18 and ab = 80, what is a² + b²? (a) 484 (b) 164 (c) 324 (d) 244
  38. 38.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1521 (b) 1681 (c) 1531 (d) 1599
  39. 39.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 4916 (b) 5476 (c) 4356 (d) 5486
  40. 40.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 32) (b) (x − 4)(x − 8) (c) (x + 1)(x + 11) (d) (x + 4)(x + 8)

Answer key

  1. 7921
  2. 484
  3. 63
  4. 30
  5. 58
  6. 25
  7. (x + 5)(x + 7)
  8. (x + 4)(x + 5)
  9. 529
  10. (x + 8)(x + 9)
  11. 109
  12. (x + 2)(x + 3)
  13. (x + 7)(x + 9)
  14. (x + 4)(x + 9)
  15. 8281
  16. 3 and 4
  17. 16
  18. (x + 5)(x + 6)
  19. 1
  20. 35
  21. 145
  22. 2916
  23. 185
  24. 576
  25. 7225
  26. 7396
  27. 27
  28. 11
  29. 3481
  30. 56
  31. (x + 3)(x + 5)
  32. 48
  33. 1
  34. 106
  35. 64
  36. (x + 2)(x + 9)
  37. 164
  38. 1521
  39. 5476
  40. (x + 4)(x + 8)

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