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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 410 · 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 + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 1)(x + 48) (c) (x + 6)(x + 8) (d) (x + 1)(x + 13)
  2. 2.What is the sum of the roots of x² − 13x + 42 = 0? (a) 42 (b) 13 (c) 12 (d) 14
  3. 3.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 4356 (b) 5486 (c) 5476 (d) 4916
  4. 4.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)
  5. 5.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 1225 (b) 2035 (c) 2025 (d) 1625
  6. 6.If a + b = 15 and ab = 50, what is (a − b)²? (a) 125 (b) 225 (c) 25 (d) 5
  7. 7.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 4901 (b) 5041 (c) 5051 (d) 4761
  8. 8.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 11025 (b) 11035 (c) 9025 (d) 10025
  9. 9.What are the roots of x² − 13x + 40 = 0? (a) 2 and 20 (b) 5 and 8 (c) 1 and 40 (d) 1 and 12
  10. 10.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 6416 (b) 5776 (c) 7056 (d) 7066
  11. 11.What are the roots of x² − 16x + 63 = 0? (a) 3 and 21 (b) 1 and 63 (c) 1 and 15 (d) 7 and 9
  12. 12.If a + b = 19 and ab = 90, what is a² + b²? (a) 541 (b) 181 (c) 361 (d) 271
  13. 13.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 451 (b) 401 (c) 441 (d) 361
  14. 14.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1849 (b) 1379 (c) 1369 (d) 1591
  15. 15.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x − 8)(x − 9) (c) (x + 1)(x + 72) (d) (x + 8)(x + 9)
  16. 16.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 961 (b) 851 (c) 899 (d) 841
  17. 17.If a + b = 19 and ab = 90, what is (a − b)²? (a) 271 (b) 1 (c) 181 (d) 361
  18. 18.What is the sum of the roots of x² − 16x + 63 = 0? (a) 17 (b) 15 (c) 63 (d) 16
  19. 19.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 5476 (b) 4366 (c) 4356 (d) 4884
  20. 20.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x − 5)(x − 6) (c) (x + 5)(x + 6) (d) (x + 1)(x + 10)
  21. 21.If a + b = 10 and ab = 21, what is a² + b²? (a) 100 (b) 79 (c) 58 (d) 142
  22. 22.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 35) (b) (x − 5)(x − 7) (c) (x + 1)(x + 11) (d) (x + 5)(x + 7)
  23. 23.Which is the correct factorisation of x² + 10x + 24? (a) (x + 4)(x + 6) (b) (x + 1)(x + 9) (c) (x − 4)(x − 6) (d) (x + 1)(x + 24)
  24. 24.What are the roots of x² − 12x + 35 = 0? (a) 5 and 7 (b) 2 and 10 (c) 1 and 11 (d) 1 and 35
  25. 25.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 14) (c) (x + 7)(x + 8) (d) (x + 1)(x + 56)
  26. 26.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 8) (b) (x + 4)(x + 5) (c) (x − 4)(x − 5) (d) (x + 1)(x + 20)
  27. 27.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10211 (b) 10001 (c) 10201 (d) 9801
  28. 28.If a + b = 12 and ab = 20, what is a² + b²? (a) 144 (b) 124 (c) 184 (d) 104
  29. 29.What are the roots of x² − 11x + 24 = 0? (a) 1 and 10 (b) 1 and 24 (c) 3 and 8 (d) 2 and 12
  30. 30.If a + b = 13 and ab = 12, what is a² + b²? (a) 193 (b) 145 (c) 157 (d) 169
  31. 31.If a + b = 11 and ab = 18, what is a² + b²? (a) 157 (b) 103 (c) 85 (d) 121
  32. 32.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 1166 (b) 676 (c) 916 (d) 1156
  33. 33.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 494 (b) 404 (c) 324 (d) 484
  34. 34.If a + b = 21 and ab = 108, what is a² + b²? (a) 441 (b) 657 (c) 333 (d) 225
  35. 35.If a + b = 11 and ab = 24, what is (a − b)²? (a) 73 (b) 5 (c) 121 (d) 25
  36. 36.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 17 (c) 18 (d) 16
  37. 37.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 1034 (b) 1024 (c) 904 (d) 784
  38. 38.If a + b = 17 and ab = 72, what is a² + b²? (a) 145 (b) 217 (c) 289 (d) 433
  39. 39.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1764 (b) 1774 (c) 1604 (d) 1444
  40. 40.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 4891 (b) 5329 (c) 4489 (d) 4499

Answer key

  1. (x + 6)(x + 8)
  2. 13
  3. 5476
  4. (x + 2)(x + 8)
  5. 2025
  6. 25
  7. 5041
  8. 11025
  9. 5 and 8
  10. 7056
  11. 7 and 9
  12. 181
  13. 441
  14. 1369
  15. (x + 8)(x + 9)
  16. 841
  17. 1
  18. 16
  19. 4356
  20. (x + 5)(x + 6)
  21. 58
  22. (x + 5)(x + 7)
  23. (x + 4)(x + 6)
  24. 5 and 7
  25. (x + 7)(x + 8)
  26. (x + 4)(x + 5)
  27. 10201
  28. 104
  29. 3 and 8
  30. 145
  31. 85
  32. 1156
  33. 484
  34. 225
  35. 25
  36. 17
  37. 1024
  38. 145
  39. 1764
  40. 4489

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