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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 545 · 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 = 14 and ab = 48, what is (a − b)²? (a) 100 (b) 4 (c) 2 (d) 196
  2. 2.If a + b = 12 and ab = 11, what is (a − b)²? (a) 144 (b) 122 (c) 10 (d) 100
  3. 3.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 9025 (b) 7225 (c) 7235 (d) 8075
  4. 4.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10009 (b) 9409 (c) 10619 (d) 10609
  5. 5.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 7921 (b) 8099 (c) 7931 (d) 8281
  6. 6.Which is the correct factorisation of x² + 10x + 21? (a) (x + 3)(x + 7) (b) (x − 3)(x − 7) (c) (x + 1)(x + 21) (d) (x + 1)(x + 9)
  7. 7.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 4916 (b) 5476 (c) 5486 (d) 4356
  8. 8.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x + 7)(x + 9) (c) (x − 7)(x − 9) (d) (x + 1)(x + 15)
  9. 9.If a + b = 11 and ab = 18, what is a² + b²? (a) 157 (b) 85 (c) 103 (d) 121
  10. 10.If a + b = 14 and ab = 24, what is (a − b)²? (a) 148 (b) 196 (c) 100 (d) 10
  11. 11.If a + b = 11 and ab = 10, what is a² + b²? (a) 101 (b) 111 (c) 121 (d) 141
  12. 12.If a + b = 12 and ab = 35, what is (a − b)²? (a) 74 (b) 144 (c) 4 (d) 2
  13. 13.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)
  14. 14.If a + b = 14 and ab = 33, what is a² + b²? (a) 130 (b) 163 (c) 262 (d) 196
  15. 15.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 8) (b) (x + 1)(x + 20) (c) (x − 4)(x − 5) (d) (x + 4)(x + 5)
  16. 16.If a + b = 11 and ab = 30, what is (a − b)²? (a) 91 (b) 1 (c) 121 (d) 61
  17. 17.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 841 (b) 901 (c) 971 (d) 961
  18. 18.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 1024 (b) 794 (c) 896 (d) 784
  19. 19.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 72 (c) 17 (d) 73
  20. 20.If a + b = 15 and ab = 56, what is (a − b)²? (a) 113 (b) 169 (c) 225 (d) 1
  21. 21.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1616 (b) 1296 (c) 1936 (d) 1946
  22. 22.If a + b = 4 and ab = 3, what is a² + b²? (a) 22 (b) 16 (c) 13 (d) 10
  23. 23.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 1225 (b) 2025 (c) 1235 (d) 1575
  24. 24.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1369 (b) 1859 (c) 1609 (d) 1849
  25. 25.Which is the correct factorisation of x² + 10x + 24? (a) (x − 4)(x − 6) (b) (x + 1)(x + 9) (c) (x + 4)(x + 6) (d) (x + 1)(x + 24)
  26. 26.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2714 (b) 2504 (c) 2704 (d) 2304
  27. 27.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)
  28. 28.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 1235 (b) 625 (c) 1225 (d) 925
  29. 29.If a + b = 13 and ab = 12, what is a² + b²? (a) 193 (b) 169 (c) 157 (d) 145
  30. 30.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 484 (b) 324 (c) 404 (d) 494
  31. 31.Which is the correct factorisation of x² + 11x + 30? (a) (x + 5)(x + 6) (b) (x − 5)(x − 6) (c) (x + 1)(x + 30) (d) (x + 1)(x + 10)
  32. 32.What is the product of the roots of x² − 11x + 24 = 0? (a) 23 (b) 25 (c) 11 (d) 24
  33. 33.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6724 (b) 6084 (c) 6094 (d) 6396
  34. 34.If a + b = 19 and ab = 90, what is (a − b)²? (a) 181 (b) 271 (c) 361 (d) 1
  35. 35.What are the roots of x² − 11x + 28 = 0? (a) 4 and 7 (b) 2 and 14 (c) 1 and 10 (d) 1 and 28
  36. 36.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2496 (b) 2304 (c) 2704 (d) 2314
  37. 37.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 5929 (b) 6899 (c) 6889 (d) 6409
  38. 38.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 4) (b) (x + 1)(x + 6) (c) (x − 2)(x − 3) (d) (x + 2)(x + 3)
  39. 39.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10001 (b) 9801 (c) 10211 (d) 10201
  40. 40.If a + b = 15 and ab = 50, what is (a − b)²? (a) 225 (b) 25 (c) 125 (d) 5

Answer key

  1. 4
  2. 100
  3. 7225
  4. 10609
  5. 7921
  6. (x + 3)(x + 7)
  7. 5476
  8. (x + 7)(x + 9)
  9. 85
  10. 100
  11. 101
  12. 4
  13. (x + 8)(x + 9)
  14. 130
  15. (x + 4)(x + 5)
  16. 1
  17. 961
  18. 784
  19. 72
  20. 1
  21. 1936
  22. 10
  23. 1225
  24. 1849
  25. (x + 4)(x + 6)
  26. 2704
  27. (x + 5)(x + 7)
  28. 1225
  29. 145
  30. 484
  31. (x + 5)(x + 6)
  32. 24
  33. 6084
  34. 1
  35. 4 and 7
  36. 2304
  37. 6889
  38. (x + 2)(x + 3)
  39. 10201
  40. 25

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