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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 37 · 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.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3364 (b) 3844 (c) 3854 (d) 3604
  2. 2.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9975 (b) 9035 (c) 9025 (d) 11025
  3. 3.If a + b = 11 and ab = 18, what is (a − b)²? (a) 121 (b) 85 (c) 49 (d) 7
  4. 4.If a + b = 5 and ab = 4, what is (a − b)²? (a) 3 (b) 9 (c) 17 (d) 25
  5. 5.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6084 (b) 6734 (c) 6724 (d) 6404
  6. 6.If a + b = 5 and ab = 6, what is (a − b)²? (a) 19 (b) 13 (c) 1 (d) 25
  7. 7.What are the roots of x² − 9x + 20 = 0? (a) 1 and 8 (b) 1 and 20 (c) 4 and 5 (d) 2 and 10
  8. 8.What is the product of the roots of x² − 15x + 56 = 0? (a) 15 (b) 57 (c) 55 (d) 56
  9. 9.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 7)(x + 8) (c) (x + 1)(x + 56) (d) (x + 1)(x + 14)
  10. 10.What is the product of the roots of x² − 7x + 12 = 0? (a) 11 (b) 13 (c) 12 (d) 7
  11. 11.What is the sum of the roots of x² − 9x + 14 = 0? (a) 8 (b) 14 (c) 10 (d) 9
  12. 12.Which is the correct factorisation of x² + 11x + 28? (a) (x + 4)(x + 7) (b) (x + 1)(x + 10) (c) (x − 4)(x − 7) (d) (x + 1)(x + 28)
  13. 13.If a + b = 15 and ab = 56, what is a² + b²? (a) 113 (b) 225 (c) 169 (d) 337
  14. 14.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2611 (b) 2601 (c) 2501 (d) 2401
  15. 15.If a + b = 12 and ab = 32, what is (a − b)²? (a) 80 (b) 4 (c) 16 (d) 144
  16. 16.What is the sum of the roots of x² − 10x + 16 = 0? (a) 16 (b) 9 (c) 11 (d) 10
  17. 17.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 16 (c) 17 (d) 18
  18. 18.If a + b = 12 and ab = 27, what is (a − b)²? (a) 144 (b) 36 (c) 6 (d) 90
  19. 19.What is the product of the roots of x² − 16x + 63 = 0? (a) 16 (b) 62 (c) 64 (d) 63
  20. 20.If a + b = 11 and ab = 30, what is (a − b)²? (a) 121 (b) 1 (c) 61 (d) 91
  21. 21.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1099 (b) 909 (c) 1089 (d) 729
  22. 22.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8291 (b) 8281 (c) 8101 (d) 7921
  23. 23.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x − 8)(x − 9) (c) (x + 1)(x + 16) (d) (x + 8)(x + 9)
  24. 24.Which is the correct factorisation of x² + 7x + 12? (a) (x + 3)(x + 4) (b) (x + 1)(x + 6) (c) (x − 3)(x − 4) (d) (x + 1)(x + 12)
  25. 25.What is the sum of the roots of x² − 7x + 10 = 0? (a) 7 (b) 6 (c) 10 (d) 8
  26. 26.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x + 1)(x + 10) (c) (x − 5)(x − 6) (d) (x + 5)(x + 6)
  27. 27.If a + b = 16 and ab = 48, what is (a − b)²? (a) 64 (b) 160 (c) 256 (d) 8
  28. 28.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1521 (b) 1691 (c) 1601 (d) 1681
  29. 29.If a + b = 9 and ab = 18, what is (a − b)²? (a) 81 (b) 3 (c) 45 (d) 9
  30. 30.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3374 (b) 3596 (c) 3364 (d) 3844
  31. 31.What are the roots of x² − 13x + 42 = 0? (a) 1 and 12 (b) 2 and 21 (c) 6 and 7 (d) 1 and 42
  32. 32.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6084 (b) 6094 (c) 6724 (d) 6396
  33. 33.If a + b = 16 and ab = 60, what is (a − b)²? (a) 256 (b) 16 (c) 4 (d) 136
  34. 34.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 32) (b) (x − 4)(x − 8) (c) (x + 1)(x + 11) (d) (x + 4)(x + 8)
  35. 35.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8474 (b) 8104 (c) 7744 (d) 8464
  36. 36.If a + b = 20 and ab = 96, what is (a − b)²? (a) 400 (b) 208 (c) 16 (d) 4
  37. 37.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 3146 (b) 3584 (c) 3136 (d) 4096
  38. 38.What is the sum of the roots of x² − 7x + 12 = 0? (a) 12 (b) 8 (c) 6 (d) 7
  39. 39.Which is the correct factorisation of x² + 12x + 35? (a) (x − 5)(x − 7) (b) (x + 1)(x + 35) (c) (x + 1)(x + 11) (d) (x + 5)(x + 7)
  40. 40.If a + b = 14 and ab = 45, what is a² + b²? (a) 106 (b) 286 (c) 151 (d) 196

Answer key

  1. 3844
  2. 9025
  3. 49
  4. 9
  5. 6724
  6. 1
  7. 4 and 5
  8. 56
  9. (x + 7)(x + 8)
  10. 12
  11. 9
  12. (x + 4)(x + 7)
  13. 113
  14. 2601
  15. 16
  16. 10
  17. 17
  18. 36
  19. 63
  20. 1
  21. 1089
  22. 8281
  23. (x + 8)(x + 9)
  24. (x + 3)(x + 4)
  25. 7
  26. (x + 5)(x + 6)
  27. 64
  28. 1681
  29. 9
  30. 3364
  31. 6 and 7
  32. 6084
  33. 16
  34. (x + 4)(x + 8)
  35. 8464
  36. 16
  37. 3136
  38. 7
  39. (x + 5)(x + 7)
  40. 106

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