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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 240 · 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 74²? (a) 5486 (b) 4916 (c) 5476 (d) 4356
  2. 2.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3721 (b) 3601 (c) 3731 (d) 3481
  3. 3.What is the sum of the roots of x² − 11x + 24 = 0? (a) 10 (b) 11 (c) 24 (d) 12
  4. 4.Which is the correct factorisation of x² + 10x + 16? (a) (x + 2)(x + 8) (b) (x − 2)(x − 8) (c) (x + 1)(x + 16) (d) (x + 1)(x + 9)
  5. 5.If a + b = 15 and ab = 36, what is a² + b²? (a) 297 (b) 225 (c) 153 (d) 189
  6. 6.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 5184 (b) 4896 (c) 4624 (d) 4634
  7. 7.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 72 (c) 16 (d) 17
  8. 8.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 21) (b) (x + 3)(x + 7) (c) (x + 1)(x + 9) (d) (x − 3)(x − 7)
  9. 9.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 8) (b) (x + 2)(x + 4) (c) (x − 2)(x − 4) (d) (x + 1)(x + 5)
  10. 10.If a + b = 10 and ab = 16, what is (a − b)²? (a) 100 (b) 68 (c) 36 (d) 6
  11. 11.What are the roots of x² − 14x + 45 = 0? (a) 3 and 15 (b) 1 and 13 (c) 1 and 45 (d) 5 and 9
  12. 12.If a + b = 11 and ab = 28, what is a² + b²? (a) 93 (b) 177 (c) 65 (d) 121
  13. 13.What is the product of the roots of x² − 9x + 18 = 0? (a) 19 (b) 18 (c) 9 (d) 17
  14. 14.If a + b = 20 and ab = 96, what is a² + b²? (a) 304 (b) 208 (c) 400 (d) 592
  15. 15.What is the sum of the roots of x² − 15x + 56 = 0? (a) 14 (b) 15 (c) 16 (d) 56
  16. 16.If a + b = 19 and ab = 88, what is (a − b)²? (a) 361 (b) 3 (c) 185 (d) 9
  17. 17.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)
  18. 18.What is the sum of the roots of x² − 9x + 20 = 0? (a) 10 (b) 9 (c) 20 (d) 8
  19. 19.If a + b = 6 and ab = 8, what is (a − b)²? (a) 2 (b) 4 (c) 20 (d) 36
  20. 20.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x − 8)(x − 9) (c) (x + 1)(x + 72) (d) (x + 8)(x + 9)
  21. 21.What is the product of the roots of x² − 10x + 16 = 0? (a) 15 (b) 16 (c) 17 (d) 10
  22. 22.What is the product of the roots of x² − 12x + 32 = 0? (a) 12 (b) 32 (c) 31 (d) 33
  23. 23.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 625 (b) 1225 (c) 925 (d) 1235
  24. 24.What is the sum of the roots of x² − 8x + 12 = 0? (a) 8 (b) 7 (c) 9 (d) 12
  25. 25.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 784 (b) 1034 (c) 904 (d) 1024
  26. 26.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 2025 (b) 1225 (c) 2035 (d) 1625
  27. 27.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 576 (b) 416 (c) 256 (d) 586
  28. 28.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 5786 (b) 5776 (c) 6384 (d) 7056
  29. 29.Which is the correct factorisation of x² + 11x + 18? (a) (x + 2)(x + 9) (b) (x + 1)(x + 10) (c) (x + 1)(x + 18) (d) (x − 2)(x − 9)
  30. 30.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x − 6)(x − 8) (c) (x + 6)(x + 8) (d) (x + 1)(x + 13)
  31. 31.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 3136 (b) 3584 (c) 4096 (d) 3146
  32. 32.If a + b = 15 and ab = 56, what is (a − b)²? (a) 225 (b) 113 (c) 169 (d) 1
  33. 33.What is the sum of the roots of x² − 12x + 35 = 0? (a) 12 (b) 11 (c) 13 (d) 35
  34. 34.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6399 (b) 6251 (c) 6561 (d) 6241
  35. 35.If a + b = 15 and ab = 56, what is a² + b²? (a) 113 (b) 169 (c) 337 (d) 225
  36. 36.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x − 7)(x − 9) (c) (x + 1)(x + 63) (d) (x + 7)(x + 9)
  37. 37.If a + b = 11 and ab = 24, what is a² + b²? (a) 97 (b) 73 (c) 169 (d) 121
  38. 38.If a + b = 16 and ab = 60, what is (a − b)²? (a) 16 (b) 4 (c) 256 (d) 136
  39. 39.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 4096 (b) 3136 (c) 4106 (d) 3616
  40. 40.If a + b = 13 and ab = 22, what is a² + b²? (a) 125 (b) 169 (c) 213 (d) 147

Answer key

  1. 5476
  2. 3721
  3. 11
  4. (x + 2)(x + 8)
  5. 153
  6. 4624
  7. 17
  8. (x + 3)(x + 7)
  9. (x + 2)(x + 4)
  10. 36
  11. 5 and 9
  12. 65
  13. 18
  14. 208
  15. 15
  16. 9
  17. (x + 4)(x + 8)
  18. 9
  19. 4
  20. (x + 8)(x + 9)
  21. 16
  22. 32
  23. 1225
  24. 8
  25. 1024
  26. 2025
  27. 576
  28. 5776
  29. (x + 2)(x + 9)
  30. (x + 6)(x + 8)
  31. 3136
  32. 1
  33. 12
  34. 6241
  35. 113
  36. (x + 7)(x + 9)
  37. 73
  38. 16
  39. 4096
  40. 125

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