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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 315 · 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.What are the roots of x² − 11x + 24 = 0? (a) 1 and 10 (b) 3 and 8 (c) 2 and 12 (d) 1 and 24
  2. 2.If a + b = 17 and ab = 72, what is (a − b)²? (a) 217 (b) 145 (c) 1 (d) 289
  3. 3.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 4925 (b) 5625 (c) 4225 (d) 5635
  4. 4.What is the sum of the roots of x² − 12x + 27 = 0? (a) 13 (b) 11 (c) 27 (d) 12
  5. 5.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)
  6. 6.If a + b = 4 and ab = 3, what is (a − b)²? (a) 10 (b) 2 (c) 16 (d) 4
  7. 7.If a + b = 9 and ab = 18, what is a² + b²? (a) 63 (b) 117 (c) 45 (d) 81
  8. 8.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 8)(x + 9) (c) (x − 8)(x − 9) (d) (x + 1)(x + 16)
  9. 9.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 10826 (b) 10016 (c) 9216 (d) 10816
  10. 10.If a + b = 14 and ab = 40, what is a² + b²? (a) 196 (b) 156 (c) 276 (d) 116
  11. 11.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 3025 (b) 4225 (c) 4235 (d) 3625
  12. 12.If a + b = 8 and ab = 7, what is (a − b)²? (a) 6 (b) 36 (c) 50 (d) 64
  13. 13.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3854 (b) 3604 (c) 3844 (d) 3364
  14. 14.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 7)(x + 9) (c) (x − 7)(x − 9) (d) (x + 1)(x + 63)
  15. 15.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x − 7)(x − 8) (c) (x + 1)(x + 56) (d) (x + 1)(x + 14)
  16. 16.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 18 (c) 16 (d) 17
  17. 17.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6401 (b) 6241 (c) 6571 (d) 6561
  18. 18.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)
  19. 19.If a + b = 21 and ab = 108, what is (a − b)²? (a) 441 (b) 3 (c) 225 (d) 9
  20. 20.What is the sum of the roots of x² − 11x + 30 = 0? (a) 10 (b) 11 (c) 12 (d) 30
  21. 21.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 28) (b) (x + 1)(x + 10) (c) (x − 4)(x − 7) (d) (x + 4)(x + 7)
  22. 22.If a + b = 14 and ab = 33, what is (a − b)²? (a) 130 (b) 196 (c) 8 (d) 64
  23. 23.What are the roots of x² − 13x + 42 = 0? (a) 2 and 21 (b) 1 and 42 (c) 6 and 7 (d) 1 and 12
  24. 24.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 8464 (b) 7754 (c) 7744 (d) 8096
  25. 25.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 4096 (b) 3584 (c) 3136 (d) 3146
  26. 26.If a + b = 13 and ab = 12, what is (a − b)²? (a) 11 (b) 145 (c) 121 (d) 169
  27. 27.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 16) (b) (x − 2)(x − 8) (c) (x + 1)(x + 9) (d) (x + 2)(x + 8)
  28. 28.What is the sum of the roots of x² − 11x + 28 = 0? (a) 10 (b) 28 (c) 12 (d) 11
  29. 29.If a + b = 18 and ab = 80, what is a² + b²? (a) 244 (b) 484 (c) 324 (d) 164
  30. 30.If a + b = 14 and ab = 48, what is (a − b)²? (a) 2 (b) 100 (c) 4 (d) 196
  31. 31.What are the roots of x² − 15x + 54 = 0? (a) 1 and 54 (b) 1 and 14 (c) 2 and 27 (d) 6 and 9
  32. 32.If a + b = 5 and ab = 4, what is (a − b)²? (a) 3 (b) 25 (c) 17 (d) 9
  33. 33.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1521 (b) 1681 (c) 1691 (d) 1601
  34. 34.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 6) (b) (x + 2)(x + 5) (c) (x + 1)(x + 10) (d) (x − 2)(x − 5)
  35. 35.If a + b = 18 and ab = 72, what is a² + b²? (a) 180 (b) 468 (c) 252 (d) 324
  36. 36.If a + b = 14 and ab = 24, what is a² + b²? (a) 196 (b) 244 (c) 148 (d) 172
  37. 37.If a + b = 17 and ab = 70, what is a² + b²? (a) 289 (b) 429 (c) 219 (d) 149
  38. 38.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 4916 (b) 5486 (c) 5476 (d) 4356
  39. 39.If a + b = 21 and ab = 108, what is a² + b²? (a) 657 (b) 333 (c) 225 (d) 441
  40. 40.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 6) (b) (x − 2)(x − 3) (c) (x + 2)(x + 3) (d) (x + 1)(x + 4)

Answer key

  1. 3 and 8
  2. 1
  3. 5625
  4. 12
  5. (x + 6)(x + 8)
  6. 4
  7. 45
  8. (x + 8)(x + 9)
  9. 10816
  10. 116
  11. 4225
  12. 36
  13. 3844
  14. (x + 7)(x + 9)
  15. (x + 7)(x + 8)
  16. 17
  17. 6561
  18. (x + 6)(x + 9)
  19. 9
  20. 11
  21. (x + 4)(x + 7)
  22. 64
  23. 6 and 7
  24. 7744
  25. 3136
  26. 121
  27. (x + 2)(x + 8)
  28. 11
  29. 164
  30. 4
  31. 6 and 9
  32. 9
  33. 1681
  34. (x + 2)(x + 5)
  35. 180
  36. 148
  37. 149
  38. 5476
  39. 225
  40. (x + 2)(x + 3)

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