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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 167 · 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 product of the roots of x² − 7x + 10 = 0? (a) 11 (b) 7 (c) 9 (d) 10
  2. 2.If a + b = 7 and ab = 10, what is (a − b)²? (a) 9 (b) 3 (c) 29 (d) 49
  3. 3.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10414 (b) 10404 (c) 10004 (d) 9604
  4. 4.Which is the correct factorisation of x² + 11x + 30? (a) (x + 5)(x + 6) (b) (x + 1)(x + 10) (c) (x + 1)(x + 30) (d) (x − 5)(x − 6)
  5. 5.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 10816 (b) 9216 (c) 9226 (d) 9984
  6. 6.What are the roots of x² − 12x + 35 = 0? (a) 5 and 7 (b) 1 and 11 (c) 1 and 35 (d) 2 and 10
  7. 7.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1521 (b) 1601 (c) 1681 (d) 1691
  8. 8.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 3146 (b) 4096 (c) 3136 (d) 3584
  9. 9.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 901 (b) 961 (c) 971 (d) 841
  10. 10.What is the sum of the roots of x² − 12x + 27 = 0? (a) 27 (b) 12 (c) 11 (d) 13
  11. 11.If a + b = 16 and ab = 63, what is a² + b²? (a) 256 (b) 130 (c) 382 (d) 193
  12. 12.What is the product of the roots of x² − 10x + 24 = 0? (a) 23 (b) 24 (c) 25 (d) 10
  13. 13.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)
  14. 14.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 289 (b) 529 (c) 409 (d) 539
  15. 15.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)
  16. 16.If a + b = 13 and ab = 22, what is a² + b²? (a) 213 (b) 125 (c) 169 (d) 147
  17. 17.If a + b = 19 and ab = 88, what is a² + b²? (a) 537 (b) 185 (c) 273 (d) 361
  18. 18.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)
  19. 19.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x + 5)(x + 7) (c) (x + 1)(x + 35) (d) (x − 5)(x − 7)
  20. 20.If a + b = 9 and ab = 14, what is a² + b²? (a) 67 (b) 109 (c) 81 (d) 53
  21. 21.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 4499 (b) 5329 (c) 4489 (d) 4891
  22. 22.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1444 (b) 1604 (c) 1774 (d) 1764
  23. 23.What is the sum of the roots of x² − 9x + 14 = 0? (a) 9 (b) 14 (c) 10 (d) 8
  24. 24.If a + b = 16 and ab = 60, what is (a − b)²? (a) 16 (b) 136 (c) 256 (d) 4
  25. 25.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6084 (b) 6404 (c) 6724 (d) 6734
  26. 26.If a + b = 11 and ab = 28, what is a² + b²? (a) 121 (b) 177 (c) 65 (d) 93
  27. 27.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 2525 (b) 3025 (c) 3035 (d) 2025
  28. 28.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 9025 (b) 8075 (c) 7225 (d) 7235
  29. 29.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 916 (b) 676 (c) 1166 (d) 1156
  30. 30.What is the sum of the roots of x² − 8x + 12 = 0? (a) 8 (b) 7 (c) 12 (d) 9
  31. 31.What is the product of the roots of x² − 13x + 36 = 0? (a) 37 (b) 36 (c) 35 (d) 13
  32. 32.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 5635 (b) 5625 (c) 4225 (d) 4925
  33. 33.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3609 (b) 3979 (c) 3249 (d) 3969
  34. 34.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 16 (c) 17 (d) 72
  35. 35.What are the roots of x² − 16x + 63 = 0? (a) 7 and 9 (b) 3 and 21 (c) 1 and 63 (d) 1 and 15
  36. 36.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 7754 (b) 8096 (c) 8464 (d) 7744
  37. 37.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 10) (b) (x − 2)(x − 5) (c) (x + 1)(x + 6) (d) (x + 2)(x + 5)
  38. 38.If a + b = 20 and ab = 99, what is (a − b)²? (a) 2 (b) 4 (c) 400 (d) 202
  39. 39.What is the product of the roots of x² − 10x + 21 = 0? (a) 22 (b) 10 (c) 20 (d) 21
  40. 40.If a + b = 10 and ab = 24, what is a² + b²? (a) 52 (b) 100 (c) 76 (d) 148

Answer key

  1. 10
  2. 9
  3. 10404
  4. (x + 5)(x + 6)
  5. 9216
  6. 5 and 7
  7. 1681
  8. 3136
  9. 961
  10. 12
  11. 130
  12. 24
  13. (x + 8)(x + 9)
  14. 529
  15. (x + 5)(x + 8)
  16. 125
  17. 185
  18. (x + 7)(x + 9)
  19. (x + 5)(x + 7)
  20. 53
  21. 4489
  22. 1764
  23. 9
  24. 16
  25. 6724
  26. 65
  27. 3025
  28. 7225
  29. 1156
  30. 8
  31. 36
  32. 5625
  33. 3969
  34. 17
  35. 7 and 9
  36. 7744
  37. (x + 2)(x + 5)
  38. 4
  39. 21
  40. 52

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