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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 226 · 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 = 16 and ab = 60, what is (a − b)²? (a) 4 (b) 16 (c) 136 (d) 256
  2. 2.If a + b = 19 and ab = 90, what is (a − b)²? (a) 271 (b) 1 (c) 181 (d) 361
  3. 3.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x + 6)(x + 8) (c) (x − 6)(x − 8) (d) (x + 1)(x + 48)
  4. 4.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 16) (c) (x + 1)(x + 72) (d) (x + 8)(x + 9)
  5. 5.If a + b = 17 and ab = 60, what is a² + b²? (a) 289 (b) 409 (c) 229 (d) 169
  6. 6.If a + b = 9 and ab = 18, what is a² + b²? (a) 81 (b) 63 (c) 45 (d) 117
  7. 7.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 8116 (b) 7396 (c) 8846 (d) 8836
  8. 8.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1604 (b) 1774 (c) 1444 (d) 1764
  9. 9.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 9226 (b) 10816 (c) 9216 (d) 9984
  10. 10.Which is the correct factorisation of x² + 14x + 45? (a) (x − 5)(x − 9) (b) (x + 5)(x + 9) (c) (x + 1)(x + 13) (d) (x + 1)(x + 45)
  11. 11.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2304 (b) 2704 (c) 2714 (d) 2504
  12. 12.What is the sum of the roots of x² − 9x + 20 = 0? (a) 8 (b) 10 (c) 9 (d) 20
  13. 13.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 17 (c) 73 (d) 72
  14. 14.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 32) (b) (x − 4)(x − 8) (c) (x + 4)(x + 8) (d) (x + 1)(x + 11)
  15. 15.Which is the correct factorisation of x² + 12x + 35? (a) (x − 5)(x − 7) (b) (x + 5)(x + 7) (c) (x + 1)(x + 35) (d) (x + 1)(x + 11)
  16. 16.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
  17. 17.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 36) (b) (x − 4)(x − 9) (c) (x + 4)(x + 9) (d) (x + 1)(x + 12)
  18. 18.If a + b = 18 and ab = 80, what is (a − b)²? (a) 4 (b) 164 (c) 324 (d) 2
  19. 19.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
  20. 20.What is the product of the roots of x² − 11x + 30 = 0? (a) 29 (b) 31 (c) 30 (d) 11
  21. 21.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 12) (b) (x + 3)(x + 4) (c) (x + 1)(x + 6) (d) (x − 3)(x − 4)
  22. 22.If a + b = 7 and ab = 6, what is a² + b²? (a) 49 (b) 43 (c) 61 (d) 37
  23. 23.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 9409 (b) 10619 (c) 10009 (d) 10609
  24. 24.What is the sum of the roots of x² − 9x + 14 = 0? (a) 9 (b) 8 (c) 14 (d) 10
  25. 25.What is the sum of the roots of x² − 13x + 36 = 0? (a) 14 (b) 13 (c) 12 (d) 36
  26. 26.If a + b = 17 and ab = 72, what is a² + b²? (a) 433 (b) 289 (c) 217 (d) 145
  27. 27.Which is the correct factorisation of x² + 9x + 18? (a) (x − 3)(x − 6) (b) (x + 1)(x + 18) (c) (x + 3)(x + 6) (d) (x + 1)(x + 8)
  28. 28.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x + 4)(x + 5) (c) (x + 1)(x + 8) (d) (x − 4)(x − 5)
  29. 29.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)
  30. 30.What is the sum of the roots of x² − 16x + 63 = 0? (a) 17 (b) 15 (c) 63 (d) 16
  31. 31.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 6) (b) (x + 2)(x + 5) (c) (x − 2)(x − 5) (d) (x + 1)(x + 10)
  32. 32.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 40) (b) (x + 1)(x + 12) (c) (x − 5)(x − 8) (d) (x + 5)(x + 8)
  33. 33.If a + b = 16 and ab = 55, what is a² + b²? (a) 366 (b) 201 (c) 256 (d) 146
  34. 34.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1849 (b) 1369 (c) 1379 (d) 1591
  35. 35.If a + b = 19 and ab = 88, what is a² + b²? (a) 537 (b) 273 (c) 185 (d) 361
  36. 36.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 3136 (b) 4106 (c) 4096 (d) 3616
  37. 37.If a + b = 10 and ab = 21, what is a² + b²? (a) 79 (b) 58 (c) 100 (d) 142
  38. 38.If a + b = 13 and ab = 30, what is (a − b)²? (a) 109 (b) 7 (c) 169 (d) 49
  39. 39.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 299 (b) 391 (c) 529 (d) 289
  40. 40.If a + b = 15 and ab = 50, what is a² + b²? (a) 325 (b) 225 (c) 125 (d) 175

Answer key

  1. 16
  2. 1
  3. (x + 6)(x + 8)
  4. (x + 8)(x + 9)
  5. 169
  6. 45
  7. 8836
  8. 1764
  9. 9216
  10. (x + 5)(x + 9)
  11. 2704
  12. 9
  13. 72
  14. (x + 4)(x + 8)
  15. (x + 5)(x + 7)
  16. 8 and 9
  17. (x + 4)(x + 9)
  18. 4
  19. 5 and 7
  20. 30
  21. (x + 3)(x + 4)
  22. 37
  23. 10609
  24. 9
  25. 13
  26. 145
  27. (x + 3)(x + 6)
  28. (x + 4)(x + 5)
  29. (x + 5)(x + 6)
  30. 16
  31. (x + 2)(x + 5)
  32. (x + 5)(x + 8)
  33. 146
  34. 1369
  35. 185
  36. 4096
  37. 58
  38. 49
  39. 289
  40. 125

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