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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 2 · 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 25²? (a) 625 (b) 1225 (c) 875 (d) 635
  2. 2.Which is the correct factorisation of x² + 11x + 28? (a) (x + 4)(x + 7) (b) (x − 4)(x − 7) (c) (x + 1)(x + 10) (d) (x + 1)(x + 28)
  3. 3.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2484 (b) 2916 (c) 2116 (d) 2126
  4. 4.If a + b = 7 and ab = 10, what is a² + b²? (a) 49 (b) 69 (c) 29 (d) 39
  5. 5.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 4356 (b) 4916 (c) 5486 (d) 5476
  6. 6.If a + b = 17 and ab = 66, what is (a − b)²? (a) 5 (b) 157 (c) 289 (d) 25
  7. 7.What is the sum of the roots of x² − 15x + 56 = 0? (a) 14 (b) 56 (c) 15 (d) 16
  8. 8.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2314 (b) 2704 (c) 2496 (d) 2304
  9. 9.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1369 (b) 1849 (c) 1859 (d) 1609
  10. 10.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 441 (b) 361 (c) 401 (d) 451
  11. 11.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 16) (c) (x + 8)(x + 9) (d) (x + 1)(x + 72)
  12. 12.What are the roots of x² − 12x + 35 = 0? (a) 5 and 7 (b) 1 and 35 (c) 1 and 11 (d) 2 and 10
  13. 13.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 324 (b) 484 (c) 334 (d) 396
  14. 14.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 8096 (b) 7754 (c) 8464 (d) 7744
  15. 15.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1531 (b) 1599 (c) 1521 (d) 1681
  16. 16.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 409 (b) 539 (c) 289 (d) 529
  17. 17.If a + b = 14 and ab = 48, what is a² + b²? (a) 100 (b) 148 (c) 196 (d) 292
  18. 18.If a + b = 7 and ab = 6, what is (a − b)²? (a) 25 (b) 49 (c) 37 (d) 5
  19. 19.What is the sum of the roots of x² − 6x + 8 = 0? (a) 5 (b) 6 (c) 8 (d) 7
  20. 20.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 7931 (b) 7921 (c) 8099 (d) 8281
  21. 21.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 5786 (b) 5776 (c) 7056 (d) 6384
  22. 22.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 1 and 72 (c) 8 and 9 (d) 2 and 36
  23. 23.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 10) (b) (x − 2)(x − 9) (c) (x + 1)(x + 18) (d) (x + 2)(x + 9)
  24. 24.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)
  25. 25.If a + b = 16 and ab = 55, what is a² + b²? (a) 256 (b) 201 (c) 146 (d) 366
  26. 26.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2401 (b) 2501 (c) 2601 (d) 2611
  27. 27.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6399 (b) 6561 (c) 6251 (d) 6241
  28. 28.If a + b = 19 and ab = 90, what is a² + b²? (a) 181 (b) 541 (c) 361 (d) 271
  29. 29.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)
  30. 30.If a + b = 16 and ab = 63, what is (a − b)²? (a) 2 (b) 130 (c) 4 (d) 256
  31. 31.If a + b = 8 and ab = 12, what is a² + b²? (a) 40 (b) 52 (c) 88 (d) 64
  32. 32.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x + 1)(x + 12) (c) (x − 6)(x − 7) (d) (x + 6)(x + 7)
  33. 33.If a + b = 8 and ab = 15, what is (a − b)²? (a) 64 (b) 4 (c) 2 (d) 34
  34. 34.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 625 (b) 1235 (c) 1225 (d) 925
  35. 35.What is the sum of the roots of x² − 9x + 18 = 0? (a) 9 (b) 8 (c) 10 (d) 18
  36. 36.If a + b = 20 and ab = 96, what is (a − b)²? (a) 16 (b) 208 (c) 4 (d) 400
  37. 37.What is the sum of the roots of x² − 8x + 12 = 0? (a) 8 (b) 12 (c) 7 (d) 9
  38. 38.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 1024 (b) 904 (c) 1034 (d) 784
  39. 39.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 12) (b) (x + 5)(x + 8) (c) (x − 5)(x − 8) (d) (x + 1)(x + 40)
  40. 40.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 10816 (b) 9226 (c) 9984 (d) 9216

Answer key

  1. 625
  2. (x + 4)(x + 7)
  3. 2116
  4. 29
  5. 5476
  6. 25
  7. 15
  8. 2304
  9. 1849
  10. 441
  11. (x + 8)(x + 9)
  12. 5 and 7
  13. 324
  14. 7744
  15. 1521
  16. 529
  17. 100
  18. 25
  19. 6
  20. 7921
  21. 5776
  22. 8 and 9
  23. (x + 2)(x + 9)
  24. (x + 4)(x + 5)
  25. 146
  26. 2601
  27. 6241
  28. 181
  29. (x + 7)(x + 8)
  30. 4
  31. 40
  32. (x + 6)(x + 7)
  33. 4
  34. 1225
  35. 9
  36. 16
  37. 8
  38. 1024
  39. (x + 5)(x + 8)
  40. 9216

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