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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 926 · 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 is the product of the roots of x² − 17x + 72 = 0? (a) 72 (b) 71 (c) 17 (d) 73
  2. 2.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1609 (b) 1369 (c) 1849 (d) 1859
  3. 3.What is the sum of the roots of x² − 14x + 45 = 0? (a) 13 (b) 15 (c) 14 (d) 45
  4. 4.If a + b = 14 and ab = 48, what is a² + b²? (a) 100 (b) 292 (c) 196 (d) 148
  5. 5.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x + 4)(x + 5) (c) (x − 4)(x − 5) (d) (x + 1)(x + 8)
  6. 6.If a + b = 10 and ab = 24, what is a² + b²? (a) 76 (b) 100 (c) 148 (d) 52
  7. 7.If a + b = 5 and ab = 6, what is (a − b)²? (a) 25 (b) 1 (c) 19 (d) 13
  8. 8.Which is the correct factorisation of x² + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 1)(x + 12) (c) (x + 3)(x + 4) (d) (x + 1)(x + 6)
  9. 9.If a + b = 11 and ab = 28, what is a² + b²? (a) 121 (b) 65 (c) 93 (d) 177
  10. 10.What is the product of the roots of x² − 9x + 18 = 0? (a) 9 (b) 19 (c) 18 (d) 17
  11. 11.Which is the correct factorisation of x² + 11x + 30? (a) (x + 5)(x + 6) (b) (x + 1)(x + 10) (c) (x + 1)(x + 30) (d) (x − 5)(x − 6)
  12. 12.What is the product of the roots of x² − 11x + 28 = 0? (a) 28 (b) 27 (c) 29 (d) 11
  13. 13.What is the sum of the roots of x² − 13x + 40 = 0? (a) 14 (b) 40 (c) 13 (d) 12
  14. 14.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1616 (b) 1946 (c) 1296 (d) 1936
  15. 15.If a + b = 17 and ab = 72, what is a² + b²? (a) 289 (b) 217 (c) 145 (d) 433
  16. 16.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1296 (b) 1306 (c) 1936 (d) 1584
  17. 17.What is the product of the roots of x² − 15x + 54 = 0? (a) 15 (b) 54 (c) 53 (d) 55
  18. 18.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 576 (b) 384 (c) 256 (d) 266
  19. 19.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 3584 (b) 3136 (c) 3146 (d) 4096
  20. 20.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 8096 (b) 8464 (c) 7744 (d) 7754
  21. 21.What is the sum of the roots of x² − 14x + 48 = 0? (a) 48 (b) 15 (c) 13 (d) 14
  22. 22.If a + b = 18 and ab = 72, what is (a − b)²? (a) 6 (b) 324 (c) 180 (d) 36
  23. 23.If a + b = 16 and ab = 60, what is a² + b²? (a) 376 (b) 256 (c) 196 (d) 136
  24. 24.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 529 (b) 299 (c) 289 (d) 391
  25. 25.Which is the correct factorisation of x² + 8x + 12? (a) (x + 2)(x + 6) (b) (x + 1)(x + 7) (c) (x − 2)(x − 6) (d) (x + 1)(x + 12)
  26. 26.If a + b = 12 and ab = 35, what is (a − b)²? (a) 2 (b) 74 (c) 4 (d) 144
  27. 27.What are the roots of x² − 10x + 21 = 0? (a) 2 and 8 (b) 3 and 7 (c) 1 and 21 (d) 1 and 9
  28. 28.Which is the correct factorisation of x² + 10x + 21? (a) (x − 3)(x − 7) (b) (x + 1)(x + 9) (c) (x + 1)(x + 21) (d) (x + 3)(x + 7)
  29. 29.Which is the correct factorisation of x² + 9x + 14? (a) (x + 2)(x + 7) (b) (x + 1)(x + 14) (c) (x + 1)(x + 8) (d) (x − 2)(x − 7)
  30. 30.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 896 (b) 794 (c) 1024 (d) 784
  31. 31.What is the product of the roots of x² − 7x + 12 = 0? (a) 7 (b) 11 (c) 12 (d) 13
  32. 32.Which is the correct factorisation of x² + 14x + 48? (a) (x + 6)(x + 8) (b) (x − 6)(x − 8) (c) (x + 1)(x + 48) (d) (x + 1)(x + 13)
  33. 33.What is the sum of the roots of x² − 12x + 35 = 0? (a) 12 (b) 35 (c) 11 (d) 13
  34. 34.If a + b = 11 and ab = 10, what is a² + b²? (a) 141 (b) 111 (c) 121 (d) 101
  35. 35.If a + b = 15 and ab = 54, what is (a − b)²? (a) 3 (b) 9 (c) 117 (d) 225
  36. 36.What is the sum of the roots of x² − 5x + 6 = 0? (a) 7 (b) 5 (c) 4 (d) 6
  37. 37.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)
  38. 38.If a + b = 20 and ab = 96, what is (a − b)²? (a) 4 (b) 400 (c) 16 (d) 208
  39. 39.What is the sum of the roots of x² − 13x + 36 = 0? (a) 14 (b) 13 (c) 36 (d) 12
  40. 40.What is the product of the roots of x² − 10x + 24 = 0? (a) 25 (b) 24 (c) 23 (d) 10

Answer key

  1. 72
  2. 1849
  3. 14
  4. 100
  5. (x + 4)(x + 5)
  6. 52
  7. 1
  8. (x + 3)(x + 4)
  9. 65
  10. 18
  11. (x + 5)(x + 6)
  12. 28
  13. 13
  14. 1936
  15. 145
  16. 1296
  17. 54
  18. 256
  19. 3136
  20. 7744
  21. 14
  22. 36
  23. 136
  24. 289
  25. (x + 2)(x + 6)
  26. 4
  27. 3 and 7
  28. (x + 3)(x + 7)
  29. (x + 2)(x + 7)
  30. 784
  31. 12
  32. (x + 6)(x + 8)
  33. 12
  34. 101
  35. 9
  36. 5
  37. (x + 8)(x + 9)
  38. 16
  39. 13
  40. 24

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