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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 4 · 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 = 22, what is (a − b)²? (a) 169 (b) 81 (c) 9 (d) 125
  2. 2.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 8649 (b) 7579 (c) 8091 (d) 7569
  3. 3.If a + b = 5 and ab = 4, what is (a − b)²? (a) 3 (b) 9 (c) 17 (d) 25
  4. 4.If a + b = 10 and ab = 24, what is (a − b)²? (a) 100 (b) 52 (c) 4 (d) 2
  5. 5.What are the roots of x² − 14x + 48 = 0? (a) 2 and 24 (b) 1 and 13 (c) 6 and 8 (d) 1 and 48
  6. 6.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)
  7. 7.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 904 (b) 1024 (c) 1034 (d) 784
  8. 8.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 9409 (b) 10009 (c) 10619 (d) 10609
  9. 9.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 6416 (b) 7056 (c) 7066 (d) 5776
  10. 10.Which is the correct factorisation of x² + 6x + 8? (a) (x − 2)(x − 4) (b) (x + 1)(x + 8) (c) (x + 1)(x + 5) (d) (x + 2)(x + 4)
  11. 11.If a + b = 10 and ab = 9, what is a² + b²? (a) 91 (b) 118 (c) 82 (d) 100
  12. 12.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)
  13. 13.If a + b = 12 and ab = 32, what is a² + b²? (a) 208 (b) 80 (c) 144 (d) 112
  14. 14.If a + b = 16 and ab = 48, what is (a − b)²? (a) 256 (b) 8 (c) 64 (d) 160
  15. 15.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2704 (b) 2714 (c) 2304 (d) 2504
  16. 16.If a + b = 9 and ab = 14, what is a² + b²? (a) 67 (b) 81 (c) 109 (d) 53
  17. 17.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 586 (b) 256 (c) 416 (d) 576
  18. 18.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)
  19. 19.What is the sum of the roots of x² − 7x + 10 = 0? (a) 10 (b) 8 (c) 6 (d) 7
  20. 20.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 2025 (b) 1225 (c) 1625 (d) 2035
  21. 21.What are the roots of x² − 15x + 56 = 0? (a) 7 and 8 (b) 1 and 14 (c) 2 and 28 (d) 1 and 56
  22. 22.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6094 (b) 6084 (c) 6724 (d) 6396
  23. 23.If a + b = 20 and ab = 96, what is a² + b²? (a) 592 (b) 304 (c) 208 (d) 400
  24. 24.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 7569 (b) 8649 (c) 8659 (d) 8109
  25. 25.Which is the correct factorisation of x² + 15x + 54? (a) (x − 6)(x − 9) (b) (x + 1)(x + 14) (c) (x + 1)(x + 54) (d) (x + 6)(x + 9)
  26. 26.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 9035 (b) 7225 (c) 8125 (d) 9025
  27. 27.What are the roots of x² − 10x + 21 = 0? (a) 1 and 9 (b) 3 and 7 (c) 2 and 8 (d) 1 and 21
  28. 28.If a + b = 8 and ab = 7, what is (a − b)²? (a) 50 (b) 64 (c) 6 (d) 36
  29. 29.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 24) (b) (x + 4)(x + 6) (c) (x − 4)(x − 6) (d) (x + 1)(x + 9)
  30. 30.What is the sum of the roots of x² − 8x + 12 = 0? (a) 8 (b) 9 (c) 12 (d) 7
  31. 31.If a + b = 14 and ab = 48, what is a² + b²? (a) 148 (b) 196 (c) 100 (d) 292
  32. 32.What is the sum of the roots of x² − 11x + 24 = 0? (a) 10 (b) 12 (c) 24 (d) 11
  33. 33.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8104 (b) 8474 (c) 8464 (d) 7744
  34. 34.If a + b = 15 and ab = 50, what is (a − b)²? (a) 5 (b) 25 (c) 125 (d) 225
  35. 35.If a + b = 15 and ab = 54, what is a² + b²? (a) 333 (b) 171 (c) 225 (d) 117
  36. 36.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 45) (b) (x + 5)(x + 9) (c) (x − 5)(x − 9) (d) (x + 1)(x + 13)
  37. 37.If a + b = 10 and ab = 21, what is a² + b²? (a) 79 (b) 58 (c) 100 (d) 142
  38. 38.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)
  39. 39.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2516 (b) 2916 (c) 2926 (d) 2116
  40. 40.If a + b = 15 and ab = 54, what is (a − b)²? (a) 225 (b) 3 (c) 117 (d) 9

Answer key

  1. 81
  2. 7569
  3. 9
  4. 4
  5. 6 and 8
  6. (x + 8)(x + 9)
  7. 1024
  8. 10609
  9. 7056
  10. (x + 2)(x + 4)
  11. 82
  12. (x + 5)(x + 7)
  13. 80
  14. 64
  15. 2704
  16. 53
  17. 576
  18. (x + 3)(x + 7)
  19. 7
  20. 2025
  21. 7 and 8
  22. 6084
  23. 208
  24. 8649
  25. (x + 6)(x + 9)
  26. 9025
  27. 3 and 7
  28. 36
  29. (x + 4)(x + 6)
  30. 8
  31. 100
  32. 11
  33. 8464
  34. 25
  35. 117
  36. (x + 5)(x + 9)
  37. 58
  38. (x + 3)(x + 6)
  39. 2916
  40. 9

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