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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 889 · 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.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 15) (b) (x + 1)(x + 7) (c) (x + 3)(x + 5) (d) (x − 3)(x − 5)
  2. 2.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 686 (b) 1156 (c) 676 (d) 884
  3. 3.What is the sum of the roots of x² − 12x + 27 = 0? (a) 13 (b) 27 (c) 11 (d) 12
  4. 4.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 635 (b) 225 (c) 625 (d) 425
  5. 5.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6251 (b) 6561 (c) 6241 (d) 6399
  6. 6.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1584 (b) 1936 (c) 1296 (d) 1306
  7. 7.If a + b = 16 and ab = 63, what is a² + b²? (a) 382 (b) 256 (c) 130 (d) 193
  8. 8.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 12) (b) (x + 5)(x + 8) (c) (x − 5)(x − 8) (d) (x + 1)(x + 40)
  9. 9.If a + b = 13 and ab = 40, what is a² + b²? (a) 169 (b) 129 (c) 249 (d) 89
  10. 10.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 13) (b) (x − 5)(x − 9) (c) (x + 1)(x + 45) (d) (x + 5)(x + 9)
  11. 11.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 1)(x + 48) (c) (x + 6)(x + 8) (d) (x + 1)(x + 13)
  12. 12.If a + b = 16 and ab = 60, what is (a − b)²? (a) 16 (b) 256 (c) 136 (d) 4
  13. 13.If a + b = 15 and ab = 44, what is (a − b)²? (a) 7 (b) 225 (c) 49 (d) 137
  14. 14.If a + b = 14 and ab = 40, what is a² + b²? (a) 276 (b) 116 (c) 196 (d) 156
  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.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 15 (c) 16 (d) 14
  17. 17.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 4499 (b) 5329 (c) 4489 (d) 4891
  18. 18.If a + b = 10 and ab = 24, what is a² + b²? (a) 148 (b) 100 (c) 76 (d) 52
  19. 19.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x + 1)(x + 72) (c) (x − 8)(x − 9) (d) (x + 1)(x + 16)
  20. 20.Which is the correct factorisation of x² + 12x + 32? (a) (x − 4)(x − 8) (b) (x + 4)(x + 8) (c) (x + 1)(x + 32) (d) (x + 1)(x + 11)
  21. 21.If a + b = 15 and ab = 36, what is a² + b²? (a) 153 (b) 225 (c) 297 (d) 189
  22. 22.What is the sum of the roots of x² − 9x + 20 = 0? (a) 10 (b) 20 (c) 9 (d) 8
  23. 23.What are the roots of x² − 10x + 21 = 0? (a) 2 and 8 (b) 1 and 21 (c) 1 and 9 (d) 3 and 7
  24. 24.If a + b = 7 and ab = 10, what is (a − b)²? (a) 49 (b) 3 (c) 9 (d) 29
  25. 25.What is the sum of the roots of x² − 12x + 32 = 0? (a) 32 (b) 13 (c) 12 (d) 11
  26. 26.If a + b = 19 and ab = 90, what is a² + b²? (a) 271 (b) 181 (c) 541 (d) 361
  27. 27.What are the roots of x² − 11x + 28 = 0? (a) 4 and 7 (b) 2 and 14 (c) 1 and 28 (d) 1 and 10
  28. 28.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 2025 (b) 2525 (c) 3025 (d) 3035
  29. 29.If a + b = 14 and ab = 45, what is a² + b²? (a) 286 (b) 196 (c) 106 (d) 151
  30. 30.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 16 (c) 17 (d) 18
  31. 31.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 5) (b) (x − 2)(x − 4) (c) (x + 2)(x + 4) (d) (x + 1)(x + 8)
  32. 32.If a + b = 4 and ab = 3, what is (a − b)²? (a) 16 (b) 10 (c) 4 (d) 2
  33. 33.If a + b = 11 and ab = 24, what is (a − b)²? (a) 121 (b) 73 (c) 5 (d) 25
  34. 34.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 73 (c) 17 (d) 72
  35. 35.What is the product of the roots of x² − 10x + 16 = 0? (a) 17 (b) 16 (c) 15 (d) 10
  36. 36.What is the product of the roots of x² − 10x + 21 = 0? (a) 10 (b) 22 (c) 21 (d) 20
  37. 37.Which is the correct factorisation of x² + 11x + 28? (a) (x + 4)(x + 7) (b) (x − 4)(x − 7) (c) (x + 1)(x + 10) (d) (x + 1)(x + 28)
  38. 38.What are the roots of x² − 13x + 42 = 0? (a) 1 and 42 (b) 6 and 7 (c) 2 and 21 (d) 1 and 12
  39. 39.Which is the correct factorisation of x² + 10x + 24? (a) (x + 4)(x + 6) (b) (x − 4)(x − 6) (c) (x + 1)(x + 9) (d) (x + 1)(x + 24)
  40. 40.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x − 5)(x − 6) (c) (x + 1)(x + 30) (d) (x + 5)(x + 6)

Answer key

  1. (x + 3)(x + 5)
  2. 676
  3. 12
  4. 625
  5. 6241
  6. 1296
  7. 130
  8. (x + 5)(x + 8)
  9. 89
  10. (x + 5)(x + 9)
  11. (x + 6)(x + 8)
  12. 16
  13. 49
  14. 116
  15. (x + 4)(x + 5)
  16. 15
  17. 4489
  18. 52
  19. (x + 8)(x + 9)
  20. (x + 4)(x + 8)
  21. 153
  22. 9
  23. 3 and 7
  24. 9
  25. 12
  26. 181
  27. 4 and 7
  28. 3025
  29. 106
  30. 17
  31. (x + 2)(x + 4)
  32. 4
  33. 25
  34. 72
  35. 16
  36. 21
  37. (x + 4)(x + 7)
  38. 6 and 7
  39. (x + 4)(x + 6)
  40. (x + 5)(x + 6)

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