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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 303 · 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 81²? (a) 6571 (b) 6561 (c) 6241 (d) 6401
  2. 2.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 784 (b) 1024 (c) 794 (d) 896
  3. 3.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)
  4. 4.Which is the correct factorisation of x² + 15x + 54? (a) (x + 6)(x + 9) (b) (x − 6)(x − 9) (c) (x + 1)(x + 14) (d) (x + 1)(x + 54)
  5. 5.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 4235 (b) 3025 (c) 3625 (d) 4225
  6. 6.If a + b = 5 and ab = 4, what is a² + b²? (a) 33 (b) 25 (c) 21 (d) 17
  7. 7.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)
  8. 8.If a + b = 9 and ab = 20, what is a² + b²? (a) 81 (b) 121 (c) 61 (d) 41
  9. 9.If a + b = 16 and ab = 48, what is (a − b)²? (a) 256 (b) 8 (c) 64 (d) 160
  10. 10.What is the sum of the roots of x² − 16x + 63 = 0? (a) 63 (b) 15 (c) 17 (d) 16
  11. 11.What is the product of the roots of x² − 16x + 63 = 0? (a) 16 (b) 62 (c) 64 (d) 63
  12. 12.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 7744 (b) 8096 (c) 8464 (d) 7754
  13. 13.If a + b = 3 and ab = 2, what is (a − b)²? (a) 1 (b) 5 (c) 9 (d) 7
  14. 14.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)
  15. 15.If a + b = 13 and ab = 12, what is a² + b²? (a) 157 (b) 193 (c) 169 (d) 145
  16. 16.What are the roots of x² − 13x + 42 = 0? (a) 1 and 12 (b) 2 and 21 (c) 6 and 7 (d) 1 and 42
  17. 17.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1859 (b) 1369 (c) 1849 (d) 1609
  18. 18.If a + b = 7 and ab = 6, what is a² + b²? (a) 61 (b) 49 (c) 37 (d) 43
  19. 19.What is the sum of the roots of x² − 11x + 30 = 0? (a) 11 (b) 12 (c) 10 (d) 30
  20. 20.If a + b = 17 and ab = 72, what is a² + b²? (a) 433 (b) 289 (c) 217 (d) 145
  21. 21.If a + b = 13 and ab = 30, what is a² + b²? (a) 139 (b) 229 (c) 169 (d) 109
  22. 22.If a + b = 19 and ab = 90, what is a² + b²? (a) 271 (b) 361 (c) 181 (d) 541
  23. 23.What is the sum of the roots of x² − 15x + 56 = 0? (a) 16 (b) 15 (c) 14 (d) 56
  24. 24.If a + b = 18 and ab = 72, what is (a − b)²? (a) 180 (b) 36 (c) 6 (d) 324
  25. 25.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 16 (c) 17 (d) 18
  26. 26.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2496 (b) 2304 (c) 2704 (d) 2314
  27. 27.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 9216 (b) 10016 (c) 10816 (d) 10826
  28. 28.If a + b = 16 and ab = 48, what is a² + b²? (a) 256 (b) 208 (c) 160 (d) 352
  29. 29.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1444 (b) 1774 (c) 1604 (d) 1764
  30. 30.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3979 (b) 3249 (c) 3609 (d) 3969
  31. 31.Which is the correct factorisation of x² + 11x + 24? (a) (x + 3)(x + 8) (b) (x − 3)(x − 8) (c) (x + 1)(x + 24) (d) (x + 1)(x + 10)
  32. 32.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 2525 (b) 2025 (c) 3035 (d) 3025
  33. 33.What are the roots of x² − 16x + 63 = 0? (a) 1 and 63 (b) 7 and 9 (c) 1 and 15 (d) 3 and 21
  34. 34.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 11) (b) (x + 3)(x + 9) (c) (x − 3)(x − 9) (d) (x + 1)(x + 27)
  35. 35.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 7066 (b) 6416 (c) 5776 (d) 7056
  36. 36.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6396 (b) 6094 (c) 6084 (d) 6724
  37. 37.If a + b = 13 and ab = 22, what is a² + b²? (a) 125 (b) 169 (c) 213 (d) 147
  38. 38.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 56) (c) (x + 1)(x + 14) (d) (x + 7)(x + 8)
  39. 39.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 1235 (b) 1225 (c) 625 (d) 925
  40. 40.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 1225 (b) 875 (c) 625 (d) 635

Answer key

  1. 6561
  2. 784
  3. (x + 8)(x + 9)
  4. (x + 6)(x + 9)
  5. 4225
  6. 17
  7. (x + 7)(x + 9)
  8. 41
  9. 64
  10. 16
  11. 63
  12. 7744
  13. 1
  14. (x + 3)(x + 5)
  15. 145
  16. 6 and 7
  17. 1849
  18. 37
  19. 11
  20. 145
  21. 109
  22. 181
  23. 15
  24. 36
  25. 17
  26. 2304
  27. 10816
  28. 160
  29. 1764
  30. 3969
  31. (x + 3)(x + 8)
  32. 3025
  33. 7 and 9
  34. (x + 3)(x + 9)
  35. 7056
  36. 6084
  37. 125
  38. (x + 7)(x + 8)
  39. 1225
  40. 625

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