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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 768 · 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.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 4925 (b) 5625 (c) 4225 (d) 5635
  2. 2.What is the product of the roots of x² − 12x + 35 = 0? (a) 36 (b) 35 (c) 34 (d) 12
  3. 3.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2809 (b) 2491 (c) 2219 (d) 2209
  4. 4.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 16 (c) 14 (d) 15
  5. 5.If a + b = 4 and ab = 3, what is (a − b)²? (a) 10 (b) 4 (c) 2 (d) 16
  6. 6.Which is the correct factorisation of x² + 10x + 21? (a) (x + 3)(x + 7) (b) (x + 1)(x + 21) (c) (x + 1)(x + 9) (d) (x − 3)(x − 7)
  7. 7.If a + b = 13 and ab = 42, what is (a − b)²? (a) 85 (b) 1 (c) 127 (d) 169
  8. 8.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 13) (b) (x + 1)(x + 45) (c) (x − 5)(x − 9) (d) (x + 5)(x + 9)
  9. 9.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)
  10. 10.If a + b = 19 and ab = 84, what is (a − b)²? (a) 25 (b) 361 (c) 5 (d) 193
  11. 11.What is the sum of the roots of x² − 16x + 63 = 0? (a) 15 (b) 16 (c) 17 (d) 63
  12. 12.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 7406 (b) 8084 (c) 7396 (d) 8836
  13. 13.Which is the correct factorisation of x² + 15x + 54? (a) (x − 6)(x − 9) (b) (x + 6)(x + 9) (c) (x + 1)(x + 54) (d) (x + 1)(x + 14)
  14. 14.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 5486 (b) 5476 (c) 4356 (d) 4916
  15. 15.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 17 (c) 72 (d) 16
  16. 16.What are the roots of x² − 13x + 42 = 0? (a) 1 and 42 (b) 2 and 21 (c) 6 and 7 (d) 1 and 12
  17. 17.Which is the correct factorisation of x² + 12x + 27? (a) (x + 3)(x + 9) (b) (x + 1)(x + 11) (c) (x + 1)(x + 27) (d) (x − 3)(x − 9)
  18. 18.If a + b = 17 and ab = 72, what is (a − b)²? (a) 289 (b) 1 (c) 217 (d) 145
  19. 19.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 409 (b) 289 (c) 539 (d) 529
  20. 20.If a + b = 16 and ab = 48, what is a² + b²? (a) 256 (b) 208 (c) 352 (d) 160
  21. 21.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 1)(x + 63) (c) (x + 7)(x + 9) (d) (x − 7)(x − 9)
  22. 22.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3979 (b) 3969 (c) 3249 (d) 3609
  23. 23.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 72 (c) 17 (d) 73
  24. 24.What is the product of the roots of x² − 13x + 36 = 0? (a) 36 (b) 37 (c) 35 (d) 13
  25. 25.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 7225 (b) 9035 (c) 9025 (d) 8125
  26. 26.If a + b = 14 and ab = 40, what is a² + b²? (a) 276 (b) 156 (c) 196 (d) 116
  27. 27.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 794 (b) 1024 (c) 784 (d) 896
  28. 28.If a + b = 3 and ab = 2, what is (a − b)²? (a) 5 (b) 1 (c) 7 (d) 9
  29. 29.If a + b = 15 and ab = 56, what is (a − b)²? (a) 1 (b) 225 (c) 169 (d) 113
  30. 30.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 1)(x + 14) (c) (x − 7)(x − 8) (d) (x + 7)(x + 8)
  31. 31.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 729 (b) 909 (c) 1099 (d) 1089
  32. 32.If a + b = 16 and ab = 60, what is a² + b²? (a) 196 (b) 136 (c) 256 (d) 376
  33. 33.Which is the correct factorisation of x² + 8x + 12? (a) (x − 2)(x − 6) (b) (x + 2)(x + 6) (c) (x + 1)(x + 12) (d) (x + 1)(x + 7)
  34. 34.If a + b = 17 and ab = 60, what is a² + b²? (a) 409 (b) 229 (c) 289 (d) 169
  35. 35.If a + b = 19 and ab = 90, what is a² + b²? (a) 541 (b) 361 (c) 181 (d) 271
  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 is the sum of the roots of x² − 8x + 15 = 0? (a) 15 (b) 9 (c) 7 (d) 8
  38. 38.If a + b = 11 and ab = 30, what is a² + b²? (a) 121 (b) 61 (c) 91 (d) 181
  39. 39.If a + b = 15 and ab = 54, what is a² + b²? (a) 333 (b) 117 (c) 225 (d) 171
  40. 40.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 10) (b) (x + 4)(x + 7) (c) (x + 1)(x + 28) (d) (x − 4)(x − 7)

Answer key

  1. 5625
  2. 35
  3. 2209
  4. 15
  5. 4
  6. (x + 3)(x + 7)
  7. 1
  8. (x + 5)(x + 9)
  9. (x + 2)(x + 5)
  10. 25
  11. 16
  12. 7396
  13. (x + 6)(x + 9)
  14. 5476
  15. 17
  16. 6 and 7
  17. (x + 3)(x + 9)
  18. 1
  19. 529
  20. 160
  21. (x + 7)(x + 9)
  22. 3969
  23. 72
  24. 36
  25. 9025
  26. 116
  27. 784
  28. 1
  29. 1
  30. (x + 7)(x + 8)
  31. 1089
  32. 136
  33. (x + 2)(x + 6)
  34. 169
  35. 181
  36. (x + 2)(x + 4)
  37. 8
  38. 61
  39. 117
  40. (x + 4)(x + 7)

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