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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 137 · 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.What is the sum of the roots of x² − 9x + 20 = 0? (a) 20 (b) 10 (c) 8 (d) 9
  2. 2.What is the sum of the roots of x² − 12x + 35 = 0? (a) 11 (b) 35 (c) 12 (d) 13
  3. 3.What are the roots of x² − 11x + 24 = 0? (a) 1 and 24 (b) 3 and 8 (c) 2 and 12 (d) 1 and 10
  4. 4.What are the roots of x² − 10x + 16 = 0? (a) 2 and 8 (b) 1 and 16 (c) 3 and 7 (d) 1 and 9
  5. 5.If a + b = 20 and ab = 99, what is a² + b²? (a) 301 (b) 400 (c) 598 (d) 202
  6. 6.What is the product of the roots of x² − 10x + 16 = 0? (a) 15 (b) 10 (c) 16 (d) 17
  7. 7.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 451 (b) 401 (c) 441 (d) 361
  8. 8.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 9991 (b) 9419 (c) 9409 (d) 10609
  9. 9.If a + b = 16 and ab = 63, what is (a − b)²? (a) 4 (b) 2 (c) 256 (d) 130
  10. 10.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1849 (b) 1609 (c) 1369 (d) 1859
  11. 11.If a + b = 15 and ab = 36, what is a² + b²? (a) 153 (b) 225 (c) 297 (d) 189
  12. 12.If a + b = 18 and ab = 77, what is (a − b)²? (a) 16 (b) 324 (c) 4 (d) 170
  13. 13.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1596 (b) 1454 (c) 1444 (d) 1764
  14. 14.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 1)(x + 14) (c) (x − 7)(x − 8) (d) (x + 7)(x + 8)
  15. 15.If a + b = 13 and ab = 40, what is a² + b²? (a) 129 (b) 89 (c) 249 (d) 169
  16. 16.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x − 5)(x − 6) (c) (x + 1)(x + 10) (d) (x + 5)(x + 6)
  17. 17.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 28) (b) (x − 4)(x − 7) (c) (x + 4)(x + 7) (d) (x + 1)(x + 10)
  18. 18.What is the sum of the roots of x² − 6x + 8 = 0? (a) 7 (b) 8 (c) 5 (d) 6
  19. 19.What are the roots of x² − 17x + 72 = 0? (a) 2 and 36 (b) 1 and 72 (c) 1 and 16 (d) 8 and 9
  20. 20.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 2475 (b) 3025 (c) 2025 (d) 2035
  21. 21.If a + b = 18 and ab = 72, what is a² + b²? (a) 468 (b) 252 (c) 324 (d) 180
  22. 22.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 35) (b) (x − 5)(x − 7) (c) (x + 5)(x + 7) (d) (x + 1)(x + 11)
  23. 23.If a + b = 10 and ab = 16, what is (a − b)²? (a) 6 (b) 36 (c) 100 (d) 68
  24. 24.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 5339 (b) 4909 (c) 4489 (d) 5329
  25. 25.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1616 (b) 1936 (c) 1296 (d) 1946
  26. 26.Which is the correct factorisation of x² + 13x + 42? (a) (x + 6)(x + 7) (b) (x + 1)(x + 12) (c) (x − 6)(x − 7) (d) (x + 1)(x + 42)
  27. 27.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 8836 (b) 8084 (c) 7396 (d) 7406
  28. 28.If a + b = 8 and ab = 12, what is a² + b²? (a) 64 (b) 40 (c) 88 (d) 52
  29. 29.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 289 (b) 539 (c) 409 (d) 529
  30. 30.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 8075 (b) 9025 (c) 7225 (d) 7235
  31. 31.Which is the correct factorisation of x² + 7x + 10? (a) (x − 2)(x − 5) (b) (x + 1)(x + 6) (c) (x + 2)(x + 5) (d) (x + 1)(x + 10)
  32. 32.If a + b = 11 and ab = 30, what is a² + b²? (a) 181 (b) 61 (c) 91 (d) 121
  33. 33.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 9409 (b) 10619 (c) 10009 (d) 10609
  34. 34.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 17 (c) 71 (d) 72
  35. 35.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 15 (c) 14 (d) 16
  36. 36.If a + b = 11 and ab = 10, what is a² + b²? (a) 141 (b) 121 (c) 101 (d) 111
  37. 37.What is the product of the roots of x² − 12x + 32 = 0? (a) 33 (b) 32 (c) 12 (d) 31
  38. 38.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4884 (b) 4366 (c) 4356 (d) 5476
  39. 39.If a + b = 3 and ab = 2, what is a² + b²? (a) 13 (b) 5 (c) 9 (d) 7
  40. 40.If a + b = 19 and ab = 90, what is a² + b²? (a) 271 (b) 181 (c) 361 (d) 541

Answer key

  1. 9
  2. 12
  3. 3 and 8
  4. 2 and 8
  5. 202
  6. 16
  7. 441
  8. 9409
  9. 4
  10. 1849
  11. 153
  12. 16
  13. 1444
  14. (x + 7)(x + 8)
  15. 89
  16. (x + 5)(x + 6)
  17. (x + 4)(x + 7)
  18. 6
  19. 8 and 9
  20. 2025
  21. 180
  22. (x + 5)(x + 7)
  23. 36
  24. 5329
  25. 1936
  26. (x + 6)(x + 7)
  27. 7396
  28. 40
  29. 529
  30. 7225
  31. (x + 2)(x + 5)
  32. 61
  33. 10609
  34. 72
  35. 15
  36. 101
  37. 32
  38. 4356
  39. 5
  40. 181

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