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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 350 · 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 = 72, what is a² + b²? (a) 433 (b) 145 (c) 289 (d) 217
  2. 2.If a + b = 21 and ab = 108, what is (a − b)²? (a) 441 (b) 3 (c) 9 (d) 225
  3. 3.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 35) (b) (x + 1)(x + 11) (c) (x + 5)(x + 7) (d) (x − 5)(x − 7)
  4. 4.What is the sum of the roots of x² − 16x + 63 = 0? (a) 15 (b) 16 (c) 63 (d) 17
  5. 5.If a + b = 10 and ab = 9, what is a² + b²? (a) 91 (b) 82 (c) 118 (d) 100
  6. 6.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8649 (b) 7569 (c) 8109 (d) 8659
  7. 7.If a + b = 16 and ab = 63, what is a² + b²? (a) 382 (b) 256 (c) 193 (d) 130
  8. 8.Which is the correct factorisation of x² + 13x + 42? (a) (x + 6)(x + 7) (b) (x − 6)(x − 7) (c) (x + 1)(x + 12) (d) (x + 1)(x + 42)
  9. 9.If a + b = 14 and ab = 40, what is a² + b²? (a) 156 (b) 196 (c) 116 (d) 276
  10. 10.What is the sum of the roots of x² − 9x + 20 = 0? (a) 9 (b) 10 (c) 8 (d) 20
  11. 11.If a + b = 19 and ab = 90, what is a² + b²? (a) 541 (b) 361 (c) 181 (d) 271
  12. 12.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 8)(x + 9) (c) (x + 1)(x + 16) (d) (x + 1)(x + 72)
  13. 13.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x − 6)(x − 8) (c) (x + 6)(x + 8) (d) (x + 1)(x + 13)
  14. 14.What is the product of the roots of x² − 14x + 45 = 0? (a) 45 (b) 44 (c) 46 (d) 14
  15. 15.If a + b = 20 and ab = 99, what is a² + b²? (a) 598 (b) 202 (c) 400 (d) 301
  16. 16.What are the roots of x² − 8x + 12 = 0? (a) 2 and 6 (b) 1 and 7 (c) 1 and 12 (d) 3 and 4
  17. 17.What is the product of the roots of x² − 15x + 56 = 0? (a) 15 (b) 57 (c) 56 (d) 55
  18. 18.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 9604 (b) 10414 (c) 10004 (d) 10404
  19. 19.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 891 (b) 729 (c) 739 (d) 1089
  20. 20.What is the sum of the roots of x² − 15x + 54 = 0? (a) 16 (b) 15 (c) 14 (d) 54
  21. 21.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2126 (b) 2916 (c) 2116 (d) 2484
  22. 22.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3591 (b) 3249 (c) 3969 (d) 3259
  23. 23.If a + b = 9 and ab = 14, what is a² + b²? (a) 81 (b) 53 (c) 67 (d) 109
  24. 24.If a + b = 13 and ab = 40, what is a² + b²? (a) 89 (b) 169 (c) 129 (d) 249
  25. 25.If a + b = 12 and ab = 35, what is a² + b²? (a) 214 (b) 74 (c) 109 (d) 144
  26. 26.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 7931 (b) 7921 (c) 8099 (d) 8281
  27. 27.If a + b = 7 and ab = 12, what is (a − b)²? (a) 25 (b) 37 (c) 49 (d) 1
  28. 28.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2809 (b) 2219 (c) 2209 (d) 2491
  29. 29.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4366 (b) 4356 (c) 4884 (d) 5476
  30. 30.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3596 (b) 3374 (c) 3364 (d) 3844
  31. 31.If a + b = 5 and ab = 6, what is a² + b²? (a) 37 (b) 19 (c) 25 (d) 13
  32. 32.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 5939 (b) 6889 (c) 5929 (d) 6391
  33. 33.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 11025 (b) 11035 (c) 10025 (d) 9025
  34. 34.If a + b = 19 and ab = 90, what is (a − b)²? (a) 271 (b) 361 (c) 181 (d) 1
  35. 35.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 12) (b) (x + 1)(x + 40) (c) (x − 5)(x − 8) (d) (x + 5)(x + 8)
  36. 36.Which is the correct factorisation of x² + 6x + 8? (a) (x − 2)(x − 4) (b) (x + 2)(x + 4) (c) (x + 1)(x + 8) (d) (x + 1)(x + 5)
  37. 37.What are the roots of x² − 7x + 10 = 0? (a) 2 and 5 (b) 1 and 10 (c) 1 and 6 (d) 3 and 4
  38. 38.What are the roots of x² − 9x + 18 = 0? (a) 3 and 6 (b) 1 and 18 (c) 2 and 9 (d) 1 and 8
  39. 39.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 4904 (b) 5194 (c) 4624 (d) 5184
  40. 40.If a + b = 13 and ab = 22, what is a² + b²? (a) 213 (b) 147 (c) 125 (d) 169

Answer key

  1. 145
  2. 9
  3. (x + 5)(x + 7)
  4. 16
  5. 82
  6. 8649
  7. 130
  8. (x + 6)(x + 7)
  9. 116
  10. 9
  11. 181
  12. (x + 8)(x + 9)
  13. (x + 6)(x + 8)
  14. 45
  15. 202
  16. 2 and 6
  17. 56
  18. 10404
  19. 729
  20. 15
  21. 2116
  22. 3249
  23. 53
  24. 89
  25. 74
  26. 7921
  27. 1
  28. 2209
  29. 4356
  30. 3364
  31. 13
  32. 5929
  33. 11025
  34. 1
  35. (x + 5)(x + 8)
  36. (x + 2)(x + 4)
  37. 2 and 5
  38. 3 and 6
  39. 5184
  40. 125

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