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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 667 · 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 = 17 and ab = 70, what is (a − b)²? (a) 149 (b) 9 (c) 3 (d) 289
  2. 2.Which is the correct factorisation of x² + 8x + 12? (a) (x − 2)(x − 6) (b) (x + 1)(x + 12) (c) (x + 2)(x + 6) (d) (x + 1)(x + 7)
  3. 3.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2401 (b) 2601 (c) 2611 (d) 2501
  4. 4.If a + b = 10 and ab = 21, what is a² + b²? (a) 79 (b) 142 (c) 58 (d) 100
  5. 5.If a + b = 16 and ab = 60, what is a² + b²? (a) 256 (b) 196 (c) 136 (d) 376
  6. 6.What are the roots of x² − 15x + 54 = 0? (a) 2 and 27 (b) 6 and 9 (c) 1 and 14 (d) 1 and 54
  7. 7.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 6391 (b) 5939 (c) 5929 (d) 6889
  8. 8.Which is the correct factorisation of x² + 11x + 30? (a) (x + 5)(x + 6) (b) (x + 1)(x + 30) (c) (x + 1)(x + 10) (d) (x − 5)(x − 6)
  9. 9.If a + b = 15 and ab = 36, what is (a − b)²? (a) 225 (b) 9 (c) 81 (d) 153
  10. 10.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)
  11. 11.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3721 (b) 3731 (c) 3481 (d) 3601
  12. 12.Which is the correct factorisation of x² + 6x + 8? (a) (x + 2)(x + 4) (b) (x + 1)(x + 8) (c) (x − 2)(x − 4) (d) (x + 1)(x + 5)
  13. 13.If a + b = 11 and ab = 24, what is a² + b²? (a) 73 (b) 121 (c) 97 (d) 169
  14. 14.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 841 (b) 901 (c) 961 (d) 971
  15. 15.If a + b = 17 and ab = 72, what is a² + b²? (a) 145 (b) 217 (c) 433 (d) 289
  16. 16.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 5329 (b) 5339 (c) 4909 (d) 4489
  17. 17.If a + b = 16 and ab = 48, what is (a − b)²? (a) 160 (b) 256 (c) 64 (d) 8
  18. 18.If a + b = 18 and ab = 72, what is (a − b)²? (a) 324 (b) 180 (c) 6 (d) 36
  19. 19.If a + b = 12 and ab = 35, what is a² + b²? (a) 214 (b) 144 (c) 74 (d) 109
  20. 20.If a + b = 19 and ab = 90, what is a² + b²? (a) 361 (b) 271 (c) 541 (d) 181
  21. 21.If a + b = 12 and ab = 32, what is a² + b²? (a) 112 (b) 208 (c) 144 (d) 80
  22. 22.If a + b = 15 and ab = 56, what is (a − b)²? (a) 169 (b) 225 (c) 1 (d) 113
  23. 23.Which is the correct factorisation of x² + 14x + 48? (a) (x + 6)(x + 8) (b) (x + 1)(x + 48) (c) (x + 1)(x + 13) (d) (x − 6)(x − 8)
  24. 24.What is the sum of the roots of x² − 15x + 56 = 0? (a) 16 (b) 56 (c) 14 (d) 15
  25. 25.If a + b = 13 and ab = 42, what is a² + b²? (a) 127 (b) 253 (c) 169 (d) 85
  26. 26.If a + b = 19 and ab = 84, what is a² + b²? (a) 529 (b) 277 (c) 361 (d) 193
  27. 27.If a + b = 10 and ab = 16, what is a² + b²? (a) 84 (b) 132 (c) 100 (d) 68
  28. 28.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10201 (b) 9801 (c) 10211 (d) 10001
  29. 29.If a + b = 18 and ab = 77, what is (a − b)²? (a) 324 (b) 170 (c) 4 (d) 16
  30. 30.What is the sum of the roots of x² − 15x + 54 = 0? (a) 15 (b) 14 (c) 54 (d) 16
  31. 31.What is the sum of the roots of x² − 11x + 30 = 0? (a) 30 (b) 10 (c) 12 (d) 11
  32. 32.If a + b = 12 and ab = 27, what is (a − b)²? (a) 90 (b) 36 (c) 144 (d) 6
  33. 33.What are the roots of x² − 12x + 35 = 0? (a) 5 and 7 (b) 2 and 10 (c) 1 and 11 (d) 1 and 35
  34. 34.If a + b = 12 and ab = 11, what is a² + b²? (a) 144 (b) 166 (c) 122 (d) 133
  35. 35.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2916 (b) 2484 (c) 2126 (d) 2116
  36. 36.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6084 (b) 6396 (c) 6724 (d) 6094
  37. 37.If a + b = 21 and ab = 108, what is (a − b)²? (a) 225 (b) 9 (c) 441 (d) 3
  38. 38.If a + b = 10 and ab = 24, what is (a − b)²? (a) 100 (b) 4 (c) 2 (d) 52
  39. 39.If a + b = 6 and ab = 8, what is a² + b²? (a) 36 (b) 20 (c) 28 (d) 52
  40. 40.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 576 (b) 266 (c) 384 (d) 256

Answer key

  1. 9
  2. (x + 2)(x + 6)
  3. 2601
  4. 58
  5. 136
  6. 6 and 9
  7. 5929
  8. (x + 5)(x + 6)
  9. 81
  10. (x + 8)(x + 9)
  11. 3721
  12. (x + 2)(x + 4)
  13. 73
  14. 961
  15. 145
  16. 5329
  17. 64
  18. 36
  19. 74
  20. 181
  21. 80
  22. 1
  23. (x + 6)(x + 8)
  24. 15
  25. 85
  26. 193
  27. 68
  28. 10201
  29. 16
  30. 15
  31. 11
  32. 36
  33. 5 and 7
  34. 122
  35. 2116
  36. 6084
  37. 9
  38. 4
  39. 20
  40. 256

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