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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 994 · 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 = 3 and ab = 2, what is (a − b)²? (a) 9 (b) 5 (c) 7 (d) 1
  2. 2.Which is the correct factorisation of x² + 7x + 10? (a) (x + 2)(x + 5) (b) (x + 1)(x + 6) (c) (x − 2)(x − 5) (d) (x + 1)(x + 10)
  3. 3.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x + 1)(x + 14) (c) (x + 6)(x + 9) (d) (x − 6)(x − 9)
  4. 4.If a + b = 21 and ab = 108, what is (a − b)²? (a) 441 (b) 9 (c) 225 (d) 3
  5. 5.If a + b = 11 and ab = 30, what is a² + b²? (a) 181 (b) 61 (c) 121 (d) 91
  6. 6.If a + b = 11 and ab = 10, what is (a − b)²? (a) 101 (b) 9 (c) 81 (d) 121
  7. 7.If a + b = 12 and ab = 27, what is (a − b)²? (a) 144 (b) 36 (c) 90 (d) 6
  8. 8.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 72 (c) 17 (d) 18
  9. 9.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 36) (b) (x − 4)(x − 9) (c) (x + 4)(x + 9) (d) (x + 1)(x + 12)
  10. 10.What is the product of the roots of x² − 14x + 45 = 0? (a) 44 (b) 45 (c) 14 (d) 46
  11. 11.If a + b = 14 and ab = 33, what is (a − b)²? (a) 196 (b) 130 (c) 64 (d) 8
  12. 12.If a + b = 18 and ab = 80, what is (a − b)²? (a) 2 (b) 164 (c) 4 (d) 324
  13. 13.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 9) (b) (x + 3)(x + 7) (c) (x + 1)(x + 21) (d) (x − 3)(x − 7)
  14. 14.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)
  15. 15.If a + b = 17 and ab = 66, what is a² + b²? (a) 289 (b) 223 (c) 157 (d) 421
  16. 16.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4366 (b) 4884 (c) 5476 (d) 4356
  17. 17.If a + b = 15 and ab = 56, what is a² + b²? (a) 169 (b) 113 (c) 337 (d) 225
  18. 18.If a + b = 13 and ab = 36, what is a² + b²? (a) 169 (b) 133 (c) 97 (d) 241
  19. 19.If a + b = 9 and ab = 14, what is (a − b)²? (a) 81 (b) 53 (c) 5 (d) 25
  20. 20.What is the sum of the roots of x² − 13x + 36 = 0? (a) 12 (b) 14 (c) 13 (d) 36
  21. 21.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 5929 (b) 6391 (c) 6889 (d) 5939
  22. 22.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 15) (b) (x − 3)(x − 5) (c) (x + 3)(x + 5) (d) (x + 1)(x + 7)
  23. 23.What is the product of the roots of x² − 11x + 30 = 0? (a) 31 (b) 30 (c) 11 (d) 29
  24. 24.If a + b = 19 and ab = 90, what is (a − b)²? (a) 181 (b) 361 (c) 1 (d) 271
  25. 25.If a + b = 15 and ab = 50, what is a² + b²? (a) 325 (b) 225 (c) 175 (d) 125
  26. 26.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x + 1)(x + 16) (c) (x − 8)(x − 9) (d) (x + 1)(x + 72)
  27. 27.If a + b = 19 and ab = 88, what is a² + b²? (a) 537 (b) 185 (c) 361 (d) 273
  28. 28.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 904 (b) 1024 (c) 784 (d) 1034
  29. 29.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 27) (b) (x + 1)(x + 11) (c) (x − 3)(x − 9) (d) (x + 3)(x + 9)
  30. 30.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 40) (b) (x + 5)(x + 8) (c) (x + 1)(x + 12) (d) (x − 5)(x − 8)
  31. 31.What is the sum of the roots of x² − 11x + 28 = 0? (a) 10 (b) 12 (c) 11 (d) 28
  32. 32.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3249 (b) 3259 (c) 3969 (d) 3591
  33. 33.What is the sum of the roots of x² − 14x + 45 = 0? (a) 13 (b) 14 (c) 45 (d) 15
  34. 34.What is the product of the roots of x² − 12x + 35 = 0? (a) 34 (b) 36 (c) 12 (d) 35
  35. 35.If a + b = 13 and ab = 22, what is a² + b²? (a) 213 (b) 125 (c) 147 (d) 169
  36. 36.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2509 (b) 2819 (c) 2809 (d) 2209
  37. 37.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 11025 (b) 9035 (c) 9025 (d) 9975
  38. 38.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8281 (b) 8291 (c) 7921 (d) 8101
  39. 39.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 3025 (b) 4225 (c) 3575 (d) 3035
  40. 40.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3599 (b) 3481 (c) 3721 (d) 3491

Answer key

  1. 1
  2. (x + 2)(x + 5)
  3. (x + 6)(x + 9)
  4. 9
  5. 61
  6. 81
  7. 36
  8. 17
  9. (x + 4)(x + 9)
  10. 45
  11. 64
  12. 4
  13. (x + 3)(x + 7)
  14. (x + 5)(x + 7)
  15. 157
  16. 4356
  17. 113
  18. 97
  19. 25
  20. 13
  21. 5929
  22. (x + 3)(x + 5)
  23. 30
  24. 1
  25. 125
  26. (x + 8)(x + 9)
  27. 185
  28. 1024
  29. (x + 3)(x + 9)
  30. (x + 5)(x + 8)
  31. 11
  32. 3249
  33. 14
  34. 35
  35. 125
  36. 2809
  37. 9025
  38. 8281
  39. 3025
  40. 3481

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