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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 26 · 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.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 1)(x + 72) (c) (x + 8)(x + 9) (d) (x − 8)(x − 9)
  2. 2.If a + b = 13 and ab = 40, what is (a − b)²? (a) 89 (b) 3 (c) 9 (d) 169
  3. 3.What is the product of the roots of x² − 14x + 48 = 0? (a) 47 (b) 14 (c) 49 (d) 48
  4. 4.If a + b = 15 and ab = 56, what is a² + b²? (a) 337 (b) 113 (c) 169 (d) 225
  5. 5.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 9) (b) (x + 4)(x + 6) (c) (x + 1)(x + 24) (d) (x − 4)(x − 6)
  6. 6.Which is the correct factorisation of x² + 9x + 20? (a) (x − 4)(x − 5) (b) (x + 1)(x + 8) (c) (x + 4)(x + 5) (d) (x + 1)(x + 20)
  7. 7.If a + b = 7 and ab = 12, what is a² + b²? (a) 25 (b) 49 (c) 37 (d) 73
  8. 8.Which is the correct factorisation of x² + 11x + 24? (a) (x − 3)(x − 8) (b) (x + 3)(x + 8) (c) (x + 1)(x + 10) (d) (x + 1)(x + 24)
  9. 9.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 1024 (b) 1034 (c) 784 (d) 904
  10. 10.What is the product of the roots of x² − 7x + 10 = 0? (a) 7 (b) 11 (c) 9 (d) 10
  11. 11.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 841 (b) 899 (c) 851 (d) 961
  12. 12.What is the product of the roots of x² − 15x + 56 = 0? (a) 57 (b) 56 (c) 15 (d) 55
  13. 13.What are the roots of x² − 15x + 56 = 0? (a) 2 and 28 (b) 7 and 8 (c) 1 and 56 (d) 1 and 14
  14. 14.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 256 (b) 586 (c) 416 (d) 576
  15. 15.If a + b = 19 and ab = 84, what is a² + b²? (a) 529 (b) 361 (c) 277 (d) 193
  16. 16.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3601 (b) 3721 (c) 3481 (d) 3731
  17. 17.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 11) (b) (x − 3)(x − 9) (c) (x + 1)(x + 27) (d) (x + 3)(x + 9)
  18. 18.If a + b = 18 and ab = 80, what is a² + b²? (a) 484 (b) 164 (c) 324 (d) 244
  19. 19.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 12) (b) (x − 2)(x − 6) (c) (x + 1)(x + 7) (d) (x + 2)(x + 6)
  20. 20.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4366 (b) 5476 (c) 4356 (d) 4884
  21. 21.If a + b = 15 and ab = 50, what is a² + b²? (a) 225 (b) 175 (c) 125 (d) 325
  22. 22.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x − 7)(x − 8) (c) (x + 1)(x + 14) (d) (x + 7)(x + 8)
  23. 23.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 8096 (b) 8464 (c) 7754 (d) 7744
  24. 24.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x + 1)(x + 42) (c) (x + 6)(x + 7) (d) (x − 6)(x − 7)
  25. 25.Which is the correct factorisation of x² + 13x + 40? (a) (x − 5)(x − 8) (b) (x + 1)(x + 12) (c) (x + 5)(x + 8) (d) (x + 1)(x + 40)
  26. 26.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 12) (b) (x − 3)(x − 4) (c) (x + 3)(x + 4) (d) (x + 1)(x + 6)
  27. 27.If a + b = 9 and ab = 14, what is (a − b)²? (a) 53 (b) 5 (c) 81 (d) 25
  28. 28.What are the roots of x² − 10x + 16 = 0? (a) 1 and 16 (b) 1 and 9 (c) 2 and 8 (d) 3 and 7
  29. 29.If a + b = 14 and ab = 24, what is a² + b²? (a) 196 (b) 172 (c) 244 (d) 148
  30. 30.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9035 (b) 9975 (c) 9025 (d) 11025
  31. 31.If a + b = 16 and ab = 55, what is a² + b²? (a) 256 (b) 366 (c) 146 (d) 201
  32. 32.Which is the correct factorisation of x² + 14x + 48? (a) (x + 6)(x + 8) (b) (x + 1)(x + 48) (c) (x − 6)(x − 8) (d) (x + 1)(x + 13)
  33. 33.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 9419 (b) 9409 (c) 9991 (d) 10609
  34. 34.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 289 (b) 391 (c) 299 (d) 529
  35. 35.If a + b = 12 and ab = 32, what is a² + b²? (a) 144 (b) 208 (c) 80 (d) 112
  36. 36.If a + b = 18 and ab = 77, what is (a − b)²? (a) 324 (b) 16 (c) 170 (d) 4
  37. 37.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 12) (b) (x + 1)(x + 36) (c) (x − 4)(x − 9) (d) (x + 4)(x + 9)
  38. 38.What are the roots of x² − 16x + 63 = 0? (a) 7 and 9 (b) 1 and 63 (c) 3 and 21 (d) 1 and 15
  39. 39.If a + b = 10 and ab = 9, what is a² + b²? (a) 82 (b) 100 (c) 118 (d) 91
  40. 40.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 1225 (b) 2025 (c) 1575 (d) 1235

Answer key

  1. (x + 8)(x + 9)
  2. 9
  3. 48
  4. 113
  5. (x + 4)(x + 6)
  6. (x + 4)(x + 5)
  7. 25
  8. (x + 3)(x + 8)
  9. 1024
  10. 10
  11. 841
  12. 56
  13. 7 and 8
  14. 576
  15. 193
  16. 3721
  17. (x + 3)(x + 9)
  18. 164
  19. (x + 2)(x + 6)
  20. 4356
  21. 125
  22. (x + 7)(x + 8)
  23. 7744
  24. (x + 6)(x + 7)
  25. (x + 5)(x + 8)
  26. (x + 3)(x + 4)
  27. 25
  28. 2 and 8
  29. 148
  30. 9025
  31. 146
  32. (x + 6)(x + 8)
  33. 9409
  34. 289
  35. 80
  36. 16
  37. (x + 4)(x + 9)
  38. 7 and 9
  39. 82
  40. 1225

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