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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 351 · 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) 4356 (b) 5486 (c) 5476 (d) 4916
  2. 2.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 7754 (b) 8096 (c) 7744 (d) 8464
  3. 3.If a + b = 19 and ab = 88, what is (a − b)²? (a) 361 (b) 185 (c) 3 (d) 9
  4. 4.If a + b = 12 and ab = 27, what is (a − b)²? (a) 90 (b) 6 (c) 36 (d) 144
  5. 5.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 971 (b) 901 (c) 961 (d) 841
  6. 6.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 6391 (b) 5939 (c) 6889 (d) 5929
  7. 7.If a + b = 6 and ab = 8, what is (a − b)²? (a) 20 (b) 36 (c) 2 (d) 4
  8. 8.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x − 8)(x − 9) (c) (x + 8)(x + 9) (d) (x + 1)(x + 72)
  9. 9.What is the product of the roots of x² − 12x + 32 = 0? (a) 33 (b) 31 (c) 32 (d) 12
  10. 10.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 7235 (b) 7225 (c) 9025 (d) 8075
  11. 11.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 14) (c) (x + 1)(x + 56) (d) (x + 7)(x + 8)
  12. 12.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 5184 (b) 4624 (c) 4896 (d) 4634
  13. 13.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 266 (b) 256 (c) 384 (d) 576
  14. 14.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 5) (b) (x − 2)(x − 4) (c) (x + 2)(x + 4) (d) (x + 1)(x + 8)
  15. 15.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3844 (b) 3596 (c) 3374 (d) 3364
  16. 16.If a + b = 17 and ab = 72, what is a² + b²? (a) 145 (b) 217 (c) 289 (d) 433
  17. 17.If a + b = 14 and ab = 45, what is (a − b)²? (a) 16 (b) 196 (c) 4 (d) 106
  18. 18.If a + b = 20 and ab = 96, what is a² + b²? (a) 208 (b) 592 (c) 400 (d) 304
  19. 19.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 11025 (b) 9975 (c) 9025 (d) 9035
  20. 20.What is the sum of the roots of x² − 12x + 35 = 0? (a) 12 (b) 13 (c) 11 (d) 35
  21. 21.If a + b = 16 and ab = 60, what is a² + b²? (a) 256 (b) 376 (c) 196 (d) 136
  22. 22.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 12) (b) (x + 1)(x + 40) (c) (x + 5)(x + 8) (d) (x − 5)(x − 8)
  23. 23.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 4) (b) (x + 2)(x + 3) (c) (x − 2)(x − 3) (d) (x + 1)(x + 6)
  24. 24.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 851 (b) 961 (c) 841 (d) 899
  25. 25.If a + b = 16 and ab = 48, what is (a − b)²? (a) 8 (b) 256 (c) 64 (d) 160
  26. 26.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 739 (b) 729 (c) 1089 (d) 891
  27. 27.If a + b = 10 and ab = 16, what is a² + b²? (a) 84 (b) 132 (c) 68 (d) 100
  28. 28.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 7235 (b) 6425 (c) 5625 (d) 7225
  29. 29.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 676 (b) 1166 (c) 916 (d) 1156
  30. 30.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9996 (b) 9614 (c) 10404 (d) 9604
  31. 31.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1099 (b) 909 (c) 1089 (d) 729
  32. 32.If a + b = 9 and ab = 8, what is (a − b)²? (a) 49 (b) 7 (c) 65 (d) 81
  33. 33.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 1024 (b) 904 (c) 1034 (d) 784
  34. 34.If a + b = 16 and ab = 63, what is a² + b²? (a) 382 (b) 130 (c) 256 (d) 193
  35. 35.What is the product of the roots of x² − 13x + 42 = 0? (a) 41 (b) 42 (c) 13 (d) 43
  36. 36.If a + b = 9 and ab = 20, what is a² + b²? (a) 81 (b) 61 (c) 121 (d) 41
  37. 37.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 7)(x + 9) (c) (x + 1)(x + 63) (d) (x + 1)(x + 15)
  38. 38.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2601 (b) 2411 (c) 2499 (d) 2401
  39. 39.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 12) (b) (x − 3)(x − 4) (c) (x + 1)(x + 6) (d) (x + 3)(x + 4)
  40. 40.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 72 (c) 16 (d) 17

Answer key

  1. 5476
  2. 7744
  3. 9
  4. 36
  5. 961
  6. 5929
  7. 4
  8. (x + 8)(x + 9)
  9. 32
  10. 7225
  11. (x + 7)(x + 8)
  12. 4624
  13. 256
  14. (x + 2)(x + 4)
  15. 3364
  16. 145
  17. 16
  18. 208
  19. 9025
  20. 12
  21. 136
  22. (x + 5)(x + 8)
  23. (x + 2)(x + 3)
  24. 841
  25. 64
  26. 729
  27. 68
  28. 7225
  29. 1156
  30. 9604
  31. 1089
  32. 49
  33. 1024
  34. 130
  35. 42
  36. 41
  37. (x + 7)(x + 9)
  38. 2401
  39. (x + 3)(x + 4)
  40. 17

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