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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 795 · 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 + 10x + 21? (a) (x + 3)(x + 7) (b) (x + 1)(x + 21) (c) (x − 3)(x − 7) (d) (x + 1)(x + 9)
  2. 2.If a + b = 18 and ab = 77, what is a² + b²? (a) 478 (b) 247 (c) 324 (d) 170
  3. 3.Which is the correct factorisation of x² + 11x + 24? (a) (x − 3)(x − 8) (b) (x + 1)(x + 24) (c) (x + 1)(x + 10) (d) (x + 3)(x + 8)
  4. 4.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2209 (b) 2819 (c) 2809 (d) 2509
  5. 5.If a + b = 7 and ab = 12, what is (a − b)²? (a) 25 (b) 49 (c) 37 (d) 1
  6. 6.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 3584 (b) 3136 (c) 4096 (d) 3146
  7. 7.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 9226 (b) 9984 (c) 9216 (d) 10816
  8. 8.What is the sum of the roots of x² − 11x + 30 = 0? (a) 30 (b) 12 (c) 11 (d) 10
  9. 9.If a + b = 10 and ab = 9, what is a² + b²? (a) 100 (b) 118 (c) 82 (d) 91
  10. 10.If a + b = 10 and ab = 9, what is (a − b)²? (a) 8 (b) 82 (c) 100 (d) 64
  11. 11.Which is the correct factorisation of x² + 13x + 36? (a) (x + 4)(x + 9) (b) (x + 1)(x + 12) (c) (x − 4)(x − 9) (d) (x + 1)(x + 36)
  12. 12.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 7406 (b) 8836 (c) 8084 (d) 7396
  13. 13.If a + b = 13 and ab = 42, what is (a − b)²? (a) 85 (b) 169 (c) 1 (d) 127
  14. 14.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1369 (b) 1859 (c) 1849 (d) 1609
  15. 15.If a + b = 14 and ab = 48, what is a² + b²? (a) 100 (b) 148 (c) 196 (d) 292
  16. 16.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x + 1)(x + 35) (c) (x + 5)(x + 7) (d) (x − 5)(x − 7)
  17. 17.What is the sum of the roots of x² − 15x + 56 = 0? (a) 16 (b) 56 (c) 14 (d) 15
  18. 18.If a + b = 18 and ab = 72, what is a² + b²? (a) 180 (b) 468 (c) 252 (d) 324
  19. 19.What is the product of the roots of x² − 11x + 30 = 0? (a) 29 (b) 31 (c) 30 (d) 11
  20. 20.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6399 (b) 6241 (c) 6561 (d) 6251
  21. 21.If a + b = 15 and ab = 36, what is a² + b²? (a) 297 (b) 189 (c) 225 (d) 153
  22. 22.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 851 (b) 899 (c) 841 (d) 961
  23. 23.If a + b = 6 and ab = 5, what is (a − b)²? (a) 26 (b) 4 (c) 16 (d) 36
  24. 24.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x − 7)(x − 9) (c) (x + 7)(x + 9) (d) (x + 1)(x + 63)
  25. 25.If a + b = 14 and ab = 40, what is (a − b)²? (a) 6 (b) 36 (c) 196 (d) 116
  26. 26.If a + b = 3 and ab = 2, what is a² + b²? (a) 13 (b) 9 (c) 5 (d) 7
  27. 27.What are the roots of x² − 11x + 30 = 0? (a) 1 and 10 (b) 2 and 15 (c) 1 and 30 (d) 5 and 6
  28. 28.What is the product of the roots of x² − 6x + 8 = 0? (a) 6 (b) 9 (c) 7 (d) 8
  29. 29.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 8281 (b) 8099 (c) 7931 (d) 7921
  30. 30.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)
  31. 31.If a + b = 10 and ab = 16, what is (a − b)²? (a) 36 (b) 68 (c) 100 (d) 6
  32. 32.If a + b = 19 and ab = 90, what is (a − b)²? (a) 271 (b) 1 (c) 361 (d) 181
  33. 33.If a + b = 18 and ab = 80, what is a² + b²? (a) 164 (b) 244 (c) 324 (d) 484
  34. 34.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 409 (b) 529 (c) 539 (d) 289
  35. 35.What is the product of the roots of x² − 9x + 18 = 0? (a) 18 (b) 9 (c) 17 (d) 19
  36. 36.If a + b = 15 and ab = 44, what is (a − b)²? (a) 49 (b) 137 (c) 7 (d) 225
  37. 37.What is the product of the roots of x² − 5x + 6 = 0? (a) 5 (b) 7 (c) 6 (d) 8
  38. 38.If a + b = 20 and ab = 99, what is a² + b²? (a) 598 (b) 301 (c) 400 (d) 202
  39. 39.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1691 (b) 1521 (c) 1681 (d) 1601
  40. 40.Which is the correct factorisation of x² + 11x + 28? (a) (x + 4)(x + 7) (b) (x − 4)(x − 7) (c) (x + 1)(x + 28) (d) (x + 1)(x + 10)

Answer key

  1. (x + 3)(x + 7)
  2. 170
  3. (x + 3)(x + 8)
  4. 2809
  5. 1
  6. 3136
  7. 9216
  8. 11
  9. 82
  10. 64
  11. (x + 4)(x + 9)
  12. 7396
  13. 1
  14. 1849
  15. 100
  16. (x + 5)(x + 7)
  17. 15
  18. 180
  19. 30
  20. 6241
  21. 153
  22. 841
  23. 16
  24. (x + 7)(x + 9)
  25. 36
  26. 5
  27. 5 and 6
  28. 8
  29. 7921
  30. (x + 8)(x + 9)
  31. 36
  32. 1
  33. 164
  34. 529
  35. 18
  36. 49
  37. 6
  38. 202
  39. 1681
  40. (x + 4)(x + 7)

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