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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 56 · 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.If a + b = 13 and ab = 36, what is a² + b²? (a) 241 (b) 133 (c) 97 (d) 169
  2. 2.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x + 1)(x + 15) (c) (x − 7)(x − 9) (d) (x + 7)(x + 9)
  3. 3.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 5339 (b) 4909 (c) 4489 (d) 5329
  4. 4.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1369 (b) 1849 (c) 1609 (d) 1859
  5. 5.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 12) (b) (x + 1)(x + 7) (c) (x − 2)(x − 6) (d) (x + 2)(x + 6)
  6. 6.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 13) (b) (x − 5)(x − 9) (c) (x + 5)(x + 9) (d) (x + 1)(x + 45)
  7. 7.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 7225 (b) 7235 (c) 6425 (d) 5625
  8. 8.Which is the correct factorisation of x² + 5x + 6? (a) (x − 2)(x − 3) (b) (x + 2)(x + 3) (c) (x + 1)(x + 4) (d) (x + 1)(x + 6)
  9. 9.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 5194 (b) 5184 (c) 4624 (d) 4904
  10. 10.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 16) (b) (x + 1)(x + 9) (c) (x + 2)(x + 8) (d) (x − 2)(x − 8)
  11. 11.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1764 (b) 1444 (c) 1774 (d) 1604
  12. 12.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 14) (c) (x + 7)(x + 8) (d) (x + 1)(x + 56)
  13. 13.Which is the correct factorisation of x² + 13x + 40? (a) (x − 5)(x − 8) (b) (x + 1)(x + 12) (c) (x + 1)(x + 40) (d) (x + 5)(x + 8)
  14. 14.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 256 (b) 416 (c) 576 (d) 586
  15. 15.If a + b = 11 and ab = 28, what is (a − b)²? (a) 9 (b) 65 (c) 3 (d) 121
  16. 16.If a + b = 16 and ab = 63, what is a² + b²? (a) 382 (b) 130 (c) 256 (d) 193
  17. 17.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 73 (c) 71 (d) 72
  18. 18.What are the roots of x² − 11x + 30 = 0? (a) 2 and 15 (b) 5 and 6 (c) 1 and 30 (d) 1 and 10
  19. 19.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1089 (b) 909 (c) 1099 (d) 729
  20. 20.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 16 (c) 17 (d) 72
  21. 21.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8659 (b) 8649 (c) 8109 (d) 7569
  22. 22.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x − 6)(x − 7) (c) (x + 1)(x + 42) (d) (x + 6)(x + 7)
  23. 23.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6251 (b) 6399 (c) 6241 (d) 6561
  24. 24.If a + b = 7 and ab = 6, what is (a − b)²? (a) 25 (b) 37 (c) 49 (d) 5
  25. 25.What is the product of the roots of x² − 13x + 42 = 0? (a) 13 (b) 43 (c) 42 (d) 41
  26. 26.If a + b = 19 and ab = 90, what is (a − b)²? (a) 181 (b) 271 (c) 361 (d) 1
  27. 27.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6724 (b) 6396 (c) 6094 (d) 6084
  28. 28.If a + b = 13 and ab = 22, what is a² + b²? (a) 169 (b) 147 (c) 213 (d) 125
  29. 29.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 1 and 16 (c) 2 and 36 (d) 8 and 9
  30. 30.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 1235 (b) 1575 (c) 1225 (d) 2025
  31. 31.If a + b = 21 and ab = 108, what is (a − b)²? (a) 225 (b) 9 (c) 3 (d) 441
  32. 32.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 9409 (b) 10619 (c) 10009 (d) 10609
  33. 33.If a + b = 18 and ab = 77, what is a² + b²? (a) 478 (b) 247 (c) 170 (d) 324
  34. 34.If a + b = 16 and ab = 55, what is a² + b²? (a) 146 (b) 201 (c) 256 (d) 366
  35. 35.If a + b = 14 and ab = 45, what is a² + b²? (a) 196 (b) 106 (c) 151 (d) 286
  36. 36.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 8099 (b) 7921 (c) 7931 (d) 8281
  37. 37.What is the product of the roots of x² − 9x + 14 = 0? (a) 14 (b) 13 (c) 15 (d) 9
  38. 38.What is the sum of the roots of x² − 10x + 16 = 0? (a) 11 (b) 16 (c) 9 (d) 10
  39. 39.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 441 (b) 361 (c) 401 (d) 451
  40. 40.If a + b = 19 and ab = 88, what is a² + b²? (a) 361 (b) 185 (c) 273 (d) 537

Answer key

  1. 97
  2. (x + 7)(x + 9)
  3. 5329
  4. 1849
  5. (x + 2)(x + 6)
  6. (x + 5)(x + 9)
  7. 7225
  8. (x + 2)(x + 3)
  9. 5184
  10. (x + 2)(x + 8)
  11. 1764
  12. (x + 7)(x + 8)
  13. (x + 5)(x + 8)
  14. 576
  15. 9
  16. 130
  17. 72
  18. 5 and 6
  19. 1089
  20. 17
  21. 8649
  22. (x + 6)(x + 7)
  23. 6241
  24. 25
  25. 42
  26. 1
  27. 6084
  28. 125
  29. 8 and 9
  30. 1225
  31. 9
  32. 10609
  33. 170
  34. 146
  35. 106
  36. 7921
  37. 14
  38. 10
  39. 441
  40. 185

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