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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 616 · 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 − 13x + 42 = 0? (a) 2 and 21 (b) 6 and 7 (c) 1 and 42 (d) 1 and 12
  2. 2.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 24) (b) (x − 3)(x − 8) (c) (x + 1)(x + 10) (d) (x + 3)(x + 8)
  3. 3.If a + b = 19 and ab = 90, what is a² + b²? (a) 271 (b) 361 (c) 541 (d) 181
  4. 4.If a + b = 13 and ab = 12, what is a² + b²? (a) 169 (b) 157 (c) 145 (d) 193
  5. 5.What is the product of the roots of x² − 15x + 56 = 0? (a) 57 (b) 56 (c) 15 (d) 55
  6. 6.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8464 (b) 8104 (c) 7744 (d) 8474
  7. 7.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2916 (b) 2126 (c) 2484 (d) 2116
  8. 8.If a + b = 21 and ab = 108, what is (a − b)²? (a) 3 (b) 225 (c) 441 (d) 9
  9. 9.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 7754 (b) 8096 (c) 7744 (d) 8464
  10. 10.If a + b = 17 and ab = 66, what is a² + b²? (a) 223 (b) 157 (c) 421 (d) 289
  11. 11.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 3146 (b) 4096 (c) 3136 (d) 3584
  12. 12.If a + b = 14 and ab = 33, what is a² + b²? (a) 163 (b) 130 (c) 262 (d) 196
  13. 13.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 45) (b) (x + 5)(x + 9) (c) (x + 1)(x + 13) (d) (x − 5)(x − 9)
  14. 14.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 10) (b) (x + 2)(x + 5) (c) (x − 2)(x − 5) (d) (x + 1)(x + 6)
  15. 15.If a + b = 16 and ab = 63, what is (a − b)²? (a) 130 (b) 2 (c) 4 (d) 256
  16. 16.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 9216 (b) 9984 (c) 9226 (d) 10816
  17. 17.If a + b = 20 and ab = 99, what is (a − b)²? (a) 4 (b) 2 (c) 202 (d) 400
  18. 18.Which is the correct factorisation of x² + 9x + 14? (a) (x − 2)(x − 7) (b) (x + 1)(x + 8) (c) (x + 2)(x + 7) (d) (x + 1)(x + 14)
  19. 19.If a + b = 13 and ab = 40, what is a² + b²? (a) 169 (b) 249 (c) 129 (d) 89
  20. 20.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2401 (b) 2601 (c) 2501 (d) 2611
  21. 21.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x − 5)(x − 7) (c) (x + 1)(x + 35) (d) (x + 5)(x + 7)
  22. 22.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6561 (b) 6241 (c) 6401 (d) 6571
  23. 23.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4366 (b) 4356 (c) 5476 (d) 4884
  24. 24.If a + b = 9 and ab = 14, what is a² + b²? (a) 53 (b) 109 (c) 67 (d) 81
  25. 25.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 2025 (b) 1235 (c) 1225 (d) 1575
  26. 26.What is the sum of the roots of x² − 15x + 54 = 0? (a) 16 (b) 15 (c) 54 (d) 14
  27. 27.If a + b = 9 and ab = 8, what is a² + b²? (a) 65 (b) 97 (c) 73 (d) 81
  28. 28.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 1)(x + 12) (c) (x + 6)(x + 7) (d) (x + 1)(x + 42)
  29. 29.What are the roots of x² − 7x + 10 = 0? (a) 3 and 4 (b) 1 and 10 (c) 1 and 6 (d) 2 and 5
  30. 30.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x + 1)(x + 56) (c) (x − 7)(x − 8) (d) (x + 1)(x + 14)
  31. 31.Which is the correct factorisation of x² + 10x + 24? (a) (x − 4)(x − 6) (b) (x + 1)(x + 24) (c) (x + 4)(x + 6) (d) (x + 1)(x + 9)
  32. 32.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 361 (b) 441 (c) 399 (d) 371
  33. 33.If a + b = 17 and ab = 66, what is (a − b)²? (a) 5 (b) 25 (c) 157 (d) 289
  34. 34.What are the roots of x² − 16x + 63 = 0? (a) 7 and 9 (b) 1 and 63 (c) 1 and 15 (d) 3 and 21
  35. 35.What are the roots of x² − 12x + 35 = 0? (a) 5 and 7 (b) 1 and 35 (c) 1 and 11 (d) 2 and 10
  36. 36.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 4925 (b) 4225 (c) 5625 (d) 5635
  37. 37.If a + b = 8 and ab = 7, what is a² + b²? (a) 57 (b) 78 (c) 64 (d) 50
  38. 38.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1691 (b) 1521 (c) 1601 (d) 1681
  39. 39.What is the product of the roots of x² − 8x + 15 = 0? (a) 8 (b) 16 (c) 15 (d) 14
  40. 40.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 8) (b) (x + 2)(x + 4) (c) (x + 1)(x + 5) (d) (x − 2)(x − 4)

Answer key

  1. 6 and 7
  2. (x + 3)(x + 8)
  3. 181
  4. 145
  5. 56
  6. 8464
  7. 2116
  8. 9
  9. 7744
  10. 157
  11. 3136
  12. 130
  13. (x + 5)(x + 9)
  14. (x + 2)(x + 5)
  15. 4
  16. 9216
  17. 4
  18. (x + 2)(x + 7)
  19. 89
  20. 2601
  21. (x + 5)(x + 7)
  22. 6561
  23. 4356
  24. 53
  25. 1225
  26. 15
  27. 65
  28. (x + 6)(x + 7)
  29. 2 and 5
  30. (x + 7)(x + 8)
  31. (x + 4)(x + 6)
  32. 361
  33. 25
  34. 7 and 9
  35. 5 and 7
  36. 5625
  37. 50
  38. 1681
  39. 15
  40. (x + 2)(x + 4)

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