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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 434 · 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 + 11x + 24? (a) (x + 3)(x + 8) (b) (x + 1)(x + 24) (c) (x − 3)(x − 8) (d) (x + 1)(x + 10)
  2. 2.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 10025 (b) 11025 (c) 11035 (d) 9025
  3. 3.If a + b = 11 and ab = 30, what is a² + b²? (a) 181 (b) 91 (c) 121 (d) 61
  4. 4.Which is the correct factorisation of x² + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 1)(x + 6) (c) (x + 1)(x + 12) (d) (x + 3)(x + 4)
  5. 5.Which is the correct factorisation of x² + 9x + 18? (a) (x + 3)(x + 6) (b) (x − 3)(x − 6) (c) (x + 1)(x + 8) (d) (x + 1)(x + 18)
  6. 6.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 3136 (b) 3584 (c) 3146 (d) 4096
  7. 7.If a + b = 17 and ab = 72, what is a² + b²? (a) 289 (b) 433 (c) 145 (d) 217
  8. 8.Which is the correct factorisation of x² + 10x + 16? (a) (x − 2)(x − 8) (b) (x + 1)(x + 9) (c) (x + 2)(x + 8) (d) (x + 1)(x + 16)
  9. 9.What is the product of the roots of x² − 15x + 56 = 0? (a) 56 (b) 55 (c) 15 (d) 57
  10. 10.What is the sum of the roots of x² − 11x + 28 = 0? (a) 11 (b) 12 (c) 10 (d) 28
  11. 11.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 7)(x + 9) (c) (x + 1)(x + 63) (d) (x − 7)(x − 9)
  12. 12.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 14) (b) (x − 2)(x − 7) (c) (x + 1)(x + 8) (d) (x + 2)(x + 7)
  13. 13.If a + b = 15 and ab = 36, what is (a − b)²? (a) 153 (b) 225 (c) 9 (d) 81
  14. 14.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 7) (b) (x + 1)(x + 15) (c) (x + 3)(x + 5) (d) (x − 3)(x − 5)
  15. 15.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 225 (b) 625 (c) 375 (d) 235
  16. 16.If a + b = 14 and ab = 40, what is (a − b)²? (a) 116 (b) 6 (c) 36 (d) 196
  17. 17.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x + 1)(x + 12) (c) (x + 6)(x + 7) (d) (x − 6)(x − 7)
  18. 18.What is the product of the roots of x² − 13x + 40 = 0? (a) 40 (b) 39 (c) 13 (d) 41
  19. 19.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2209 (b) 2809 (c) 2509 (d) 2819
  20. 20.If a + b = 14 and ab = 48, what is (a − b)²? (a) 196 (b) 2 (c) 100 (d) 4
  21. 21.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 7921 (b) 8281 (c) 8099 (d) 7931
  22. 22.Which is the correct factorisation of x² + 12x + 27? (a) (x + 3)(x + 9) (b) (x + 1)(x + 11) (c) (x − 3)(x − 9) (d) (x + 1)(x + 27)
  23. 23.What is the product of the roots of x² − 11x + 30 = 0? (a) 11 (b) 29 (c) 31 (d) 30
  24. 24.What is the sum of the roots of x² − 6x + 8 = 0? (a) 8 (b) 6 (c) 7 (d) 5
  25. 25.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 8) (b) (x + 1)(x + 5) (c) (x − 2)(x − 4) (d) (x + 2)(x + 4)
  26. 26.If a + b = 11 and ab = 10, what is a² + b²? (a) 141 (b) 101 (c) 111 (d) 121
  27. 27.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6241 (b) 6401 (c) 6561 (d) 6571
  28. 28.If a + b = 8 and ab = 12, what is (a − b)²? (a) 64 (b) 40 (c) 16 (d) 4
  29. 29.If a + b = 3 and ab = 2, what is (a − b)²? (a) 7 (b) 5 (c) 1 (d) 9
  30. 30.If a + b = 13 and ab = 40, what is (a − b)²? (a) 3 (b) 169 (c) 89 (d) 9
  31. 31.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1099 (b) 909 (c) 1089 (d) 729
  32. 32.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x + 1)(x + 56) (c) (x − 7)(x − 8) (d) (x + 7)(x + 8)
  33. 33.If a + b = 19 and ab = 84, what is (a − b)²? (a) 193 (b) 361 (c) 5 (d) 25
  34. 34.If a + b = 12 and ab = 32, what is a² + b²? (a) 80 (b) 144 (c) 112 (d) 208
  35. 35.If a + b = 16 and ab = 63, what is a² + b²? (a) 382 (b) 256 (c) 130 (d) 193
  36. 36.If a + b = 18 and ab = 77, what is (a − b)²? (a) 170 (b) 16 (c) 4 (d) 324
  37. 37.What are the roots of x² − 13x + 42 = 0? (a) 2 and 21 (b) 1 and 12 (c) 1 and 42 (d) 6 and 7
  38. 38.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 5625 (b) 4225 (c) 5635 (d) 4925
  39. 39.What is the product of the roots of x² − 12x + 32 = 0? (a) 12 (b) 31 (c) 33 (d) 32
  40. 40.If a + b = 16 and ab = 60, what is (a − b)²? (a) 16 (b) 256 (c) 136 (d) 4

Answer key

  1. (x + 3)(x + 8)
  2. 11025
  3. 61
  4. (x + 3)(x + 4)
  5. (x + 3)(x + 6)
  6. 3136
  7. 145
  8. (x + 2)(x + 8)
  9. 56
  10. 11
  11. (x + 7)(x + 9)
  12. (x + 2)(x + 7)
  13. 81
  14. (x + 3)(x + 5)
  15. 225
  16. 36
  17. (x + 6)(x + 7)
  18. 40
  19. 2809
  20. 4
  21. 7921
  22. (x + 3)(x + 9)
  23. 30
  24. 6
  25. (x + 2)(x + 4)
  26. 101
  27. 6561
  28. 16
  29. 1
  30. 9
  31. 1089
  32. (x + 7)(x + 8)
  33. 25
  34. 80
  35. 130
  36. 16
  37. 6 and 7
  38. 5625
  39. 32
  40. 16

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