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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 537 · 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 = 40, what is a² + b²? (a) 129 (b) 169 (c) 89 (d) 249
  2. 2.If a + b = 11 and ab = 28, what is a² + b²? (a) 121 (b) 93 (c) 177 (d) 65
  3. 3.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 416 (b) 256 (c) 586 (d) 576
  4. 4.If a + b = 20 and ab = 96, what is (a − b)²? (a) 208 (b) 16 (c) 400 (d) 4
  5. 5.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 7) (b) (x − 2)(x − 6) (c) (x + 1)(x + 12) (d) (x + 2)(x + 6)
  6. 6.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3601 (b) 3481 (c) 3731 (d) 3721
  7. 7.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 10) (b) (x + 1)(x + 28) (c) (x − 4)(x − 7) (d) (x + 4)(x + 7)
  8. 8.If a + b = 16 and ab = 55, what is a² + b²? (a) 366 (b) 256 (c) 146 (d) 201
  9. 9.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 10816 (b) 9216 (c) 9226 (d) 9984
  10. 10.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x + 1)(x + 14) (c) (x + 1)(x + 56) (d) (x − 7)(x − 8)
  11. 11.If a + b = 15 and ab = 36, what is (a − b)²? (a) 225 (b) 81 (c) 153 (d) 9
  12. 12.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 17 (c) 18 (d) 72
  13. 13.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2116 (b) 2126 (c) 2484 (d) 2916
  14. 14.If a + b = 19 and ab = 88, what is (a − b)²? (a) 185 (b) 361 (c) 3 (d) 9
  15. 15.If a + b = 4 and ab = 3, what is a² + b²? (a) 22 (b) 16 (c) 13 (d) 10
  16. 16.What is the product of the roots of x² − 12x + 35 = 0? (a) 36 (b) 35 (c) 34 (d) 12
  17. 17.If a + b = 9 and ab = 20, what is (a − b)²? (a) 1 (b) 61 (c) 81 (d) 41
  18. 18.If a + b = 17 and ab = 60, what is (a − b)²? (a) 289 (b) 7 (c) 49 (d) 169
  19. 19.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x − 6)(x − 9) (c) (x + 6)(x + 9) (d) (x + 1)(x + 14)
  20. 20.If a + b = 20 and ab = 99, what is a² + b²? (a) 202 (b) 598 (c) 400 (d) 301
  21. 21.If a + b = 19 and ab = 88, what is a² + b²? (a) 537 (b) 185 (c) 273 (d) 361
  22. 22.Which is the correct factorisation of x² + 7x + 10? (a) (x + 2)(x + 5) (b) (x − 2)(x − 5) (c) (x + 1)(x + 6) (d) (x + 1)(x + 10)
  23. 23.What is the sum of the roots of x² − 11x + 28 = 0? (a) 12 (b) 11 (c) 28 (d) 10
  24. 24.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 2 and 36 (c) 1 and 72 (d) 8 and 9
  25. 25.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 4909 (b) 4489 (c) 5329 (d) 5339
  26. 26.If a + b = 16 and ab = 60, what is a² + b²? (a) 196 (b) 256 (c) 136 (d) 376
  27. 27.If a + b = 17 and ab = 70, what is a² + b²? (a) 289 (b) 429 (c) 149 (d) 219
  28. 28.What are the roots of x² − 15x + 54 = 0? (a) 1 and 54 (b) 6 and 9 (c) 1 and 14 (d) 2 and 27
  29. 29.What is the product of the roots of x² − 6x + 8 = 0? (a) 7 (b) 9 (c) 8 (d) 6
  30. 30.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x − 6)(x − 7) (c) (x + 1)(x + 12) (d) (x + 6)(x + 7)
  31. 31.If a + b = 21 and ab = 108, what is a² + b²? (a) 333 (b) 441 (c) 225 (d) 657
  32. 32.If a + b = 15 and ab = 54, what is a² + b²? (a) 333 (b) 171 (c) 225 (d) 117
  33. 33.If a + b = 12 and ab = 20, what is (a − b)²? (a) 104 (b) 64 (c) 144 (d) 8
  34. 34.If a + b = 11 and ab = 24, what is a² + b²? (a) 97 (b) 73 (c) 121 (d) 169
  35. 35.What is the sum of the roots of x² − 13x + 42 = 0? (a) 42 (b) 14 (c) 13 (d) 12
  36. 36.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 7235 (b) 5625 (c) 7225 (d) 6425
  37. 37.If a + b = 12 and ab = 35, what is a² + b²? (a) 144 (b) 109 (c) 214 (d) 74
  38. 38.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 10404 (b) 9604 (c) 9614 (d) 9996
  39. 39.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3249 (b) 3591 (c) 3969 (d) 3259
  40. 40.If a + b = 13 and ab = 36, what is (a − b)²? (a) 5 (b) 25 (c) 169 (d) 97

Answer key

  1. 89
  2. 65
  3. 576
  4. 16
  5. (x + 2)(x + 6)
  6. 3721
  7. (x + 4)(x + 7)
  8. 146
  9. 9216
  10. (x + 7)(x + 8)
  11. 81
  12. 17
  13. 2116
  14. 9
  15. 10
  16. 35
  17. 1
  18. 49
  19. (x + 6)(x + 9)
  20. 202
  21. 185
  22. (x + 2)(x + 5)
  23. 11
  24. 8 and 9
  25. 5329
  26. 136
  27. 149
  28. 6 and 9
  29. 8
  30. (x + 6)(x + 7)
  31. 225
  32. 117
  33. 64
  34. 73
  35. 13
  36. 7225
  37. 74
  38. 9604
  39. 3249
  40. 25

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