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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 476 · 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.If a + b = 17 and ab = 72, what is a² + b²? (a) 433 (b) 217 (c) 289 (d) 145
  2. 2.If a + b = 16 and ab = 55, what is (a − b)²? (a) 6 (b) 146 (c) 36 (d) 256
  3. 3.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10001 (b) 9801 (c) 10201 (d) 10211
  4. 4.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 7744 (b) 8464 (c) 8104 (d) 8474
  5. 5.If a + b = 13 and ab = 40, what is a² + b²? (a) 249 (b) 169 (c) 129 (d) 89
  6. 6.If a + b = 5 and ab = 4, what is (a − b)²? (a) 9 (b) 3 (c) 25 (d) 17
  7. 7.If a + b = 7 and ab = 6, what is (a − b)²? (a) 49 (b) 37 (c) 25 (d) 5
  8. 8.If a + b = 17 and ab = 60, what is (a − b)²? (a) 49 (b) 169 (c) 289 (d) 7
  9. 9.What are the roots of x² − 15x + 56 = 0? (a) 1 and 14 (b) 1 and 56 (c) 2 and 28 (d) 7 and 8
  10. 10.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 635 (b) 225 (c) 625 (d) 425
  11. 11.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 14) (c) (x + 7)(x + 8) (d) (x + 1)(x + 56)
  12. 12.What is the sum of the roots of x² − 13x + 40 = 0? (a) 14 (b) 12 (c) 40 (d) 13
  13. 13.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 7396 (b) 8116 (c) 8836 (d) 8846
  14. 14.If a + b = 15 and ab = 50, what is a² + b²? (a) 225 (b) 325 (c) 175 (d) 125
  15. 15.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 21) (b) (x + 1)(x + 9) (c) (x − 3)(x − 7) (d) (x + 3)(x + 7)
  16. 16.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 5776 (b) 7056 (c) 5786 (d) 6384
  17. 17.What are the roots of x² − 9x + 18 = 0? (a) 3 and 6 (b) 2 and 9 (c) 1 and 18 (d) 1 and 8
  18. 18.If a + b = 13 and ab = 30, what is (a − b)²? (a) 169 (b) 7 (c) 109 (d) 49
  19. 19.What are the roots of x² − 6x + 8 = 0? (a) 2 and 5 (b) 1 and 5 (c) 2 and 4 (d) 1 and 8
  20. 20.If a + b = 17 and ab = 70, what is (a − b)²? (a) 149 (b) 289 (c) 3 (d) 9
  21. 21.If a + b = 3 and ab = 2, what is a² + b²? (a) 9 (b) 13 (c) 7 (d) 5
  22. 22.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x + 1)(x + 10) (c) (x − 5)(x − 6) (d) (x + 5)(x + 6)
  23. 23.If a + b = 13 and ab = 42, what is a² + b²? (a) 169 (b) 253 (c) 127 (d) 85
  24. 24.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1849 (b) 1591 (c) 1369 (d) 1379
  25. 25.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)
  26. 26.What are the roots of x² − 14x + 48 = 0? (a) 6 and 8 (b) 1 and 48 (c) 1 and 13 (d) 2 and 24
  27. 27.What is the product of the roots of x² − 8x + 12 = 0? (a) 13 (b) 11 (c) 8 (d) 12
  28. 28.If a + b = 17 and ab = 60, what is a² + b²? (a) 169 (b) 289 (c) 409 (d) 229
  29. 29.If a + b = 16 and ab = 63, what is a² + b²? (a) 193 (b) 256 (c) 130 (d) 382
  30. 30.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 10) (b) (x + 1)(x + 6) (c) (x + 2)(x + 5) (d) (x − 2)(x − 5)
  31. 31.If a + b = 9 and ab = 18, what is (a − b)²? (a) 9 (b) 45 (c) 3 (d) 81
  32. 32.If a + b = 12 and ab = 20, what is (a − b)²? (a) 8 (b) 64 (c) 144 (d) 104
  33. 33.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 12) (b) (x − 2)(x − 6) (c) (x + 2)(x + 6) (d) (x + 1)(x + 7)
  34. 34.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 1166 (b) 1156 (c) 676 (d) 916
  35. 35.If a + b = 10 and ab = 21, what is (a − b)²? (a) 58 (b) 16 (c) 100 (d) 4
  36. 36.What is the product of the roots of x² − 13x + 40 = 0? (a) 13 (b) 40 (c) 41 (d) 39
  37. 37.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 739 (b) 891 (c) 1089 (d) 729
  38. 38.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 371 (b) 361 (c) 399 (d) 441
  39. 39.If a + b = 6 and ab = 8, what is (a − b)²? (a) 4 (b) 36 (c) 2 (d) 20
  40. 40.If a + b = 11 and ab = 18, what is (a − b)²? (a) 85 (b) 121 (c) 49 (d) 7

Answer key

  1. 145
  2. 36
  3. 10201
  4. 8464
  5. 89
  6. 9
  7. 25
  8. 49
  9. 7 and 8
  10. 625
  11. (x + 7)(x + 8)
  12. 13
  13. 8836
  14. 125
  15. (x + 3)(x + 7)
  16. 5776
  17. 3 and 6
  18. 49
  19. 2 and 4
  20. 9
  21. 5
  22. (x + 5)(x + 6)
  23. 85
  24. 1369
  25. (x + 8)(x + 9)
  26. 6 and 8
  27. 12
  28. 169
  29. 130
  30. (x + 2)(x + 5)
  31. 9
  32. 64
  33. (x + 2)(x + 6)
  34. 1156
  35. 16
  36. 40
  37. 729
  38. 361
  39. 4
  40. 49

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