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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 626 · 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.If a + b = 5 and ab = 6, what is (a − b)²? (a) 1 (b) 19 (c) 25 (d) 13
  2. 2.If a + b = 10 and ab = 16, what is a² + b²? (a) 132 (b) 100 (c) 84 (d) 68
  3. 3.If a + b = 20 and ab = 96, what is a² + b²? (a) 592 (b) 400 (c) 208 (d) 304
  4. 4.What are the roots of x² − 7x + 10 = 0? (a) 1 and 10 (b) 3 and 4 (c) 2 and 5 (d) 1 and 6
  5. 5.If a + b = 4 and ab = 3, what is (a − b)²? (a) 2 (b) 16 (c) 4 (d) 10
  6. 6.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 9025 (b) 8075 (c) 7235 (d) 7225
  7. 7.If a + b = 13 and ab = 22, what is a² + b²? (a) 213 (b) 125 (c) 169 (d) 147
  8. 8.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1296 (b) 1936 (c) 1306 (d) 1584
  9. 9.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)
  10. 10.Which is the correct factorisation of x² + 9x + 20? (a) (x + 4)(x + 5) (b) (x − 4)(x − 5) (c) (x + 1)(x + 8) (d) (x + 1)(x + 20)
  11. 11.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4899 (b) 5041 (c) 4771 (d) 4761
  12. 12.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x + 6)(x + 7) (c) (x − 6)(x − 7) (d) (x + 1)(x + 12)
  13. 13.If a + b = 17 and ab = 66, what is (a − b)²? (a) 289 (b) 5 (c) 25 (d) 157
  14. 14.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 625 (b) 1235 (c) 1225 (d) 925
  15. 15.Which is the correct factorisation of x² + 9x + 18? (a) (x − 3)(x − 6) (b) (x + 1)(x + 18) (c) (x + 3)(x + 6) (d) (x + 1)(x + 8)
  16. 16.What is the product of the roots of x² − 15x + 54 = 0? (a) 55 (b) 54 (c) 53 (d) 15
  17. 17.If a + b = 8 and ab = 12, what is (a − b)²? (a) 64 (b) 16 (c) 40 (d) 4
  18. 18.What is the sum of the roots of x² − 10x + 24 = 0? (a) 10 (b) 11 (c) 24 (d) 9
  19. 19.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1379 (b) 1591 (c) 1369 (d) 1849
  20. 20.If a + b = 7 and ab = 12, what is a² + b²? (a) 37 (b) 25 (c) 73 (d) 49
  21. 21.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 391 (b) 299 (c) 289 (d) 529
  22. 22.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)
  23. 23.Which is the correct factorisation of x² + 12x + 32? (a) (x − 4)(x − 8) (b) (x + 1)(x + 11) (c) (x + 4)(x + 8) (d) (x + 1)(x + 32)
  24. 24.If a + b = 6 and ab = 8, what is a² + b²? (a) 28 (b) 20 (c) 52 (d) 36
  25. 25.If a + b = 6 and ab = 5, what is a² + b²? (a) 26 (b) 31 (c) 36 (d) 46
  26. 26.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2516 (b) 2116 (c) 2926 (d) 2916
  27. 27.What is the product of the roots of x² − 9x + 20 = 0? (a) 19 (b) 9 (c) 20 (d) 21
  28. 28.If a + b = 18 and ab = 72, what is (a − b)²? (a) 6 (b) 180 (c) 324 (d) 36
  29. 29.If a + b = 8 and ab = 12, what is a² + b²? (a) 88 (b) 64 (c) 40 (d) 52
  30. 30.What is the sum of the roots of x² − 13x + 36 = 0? (a) 36 (b) 12 (c) 13 (d) 14
  31. 31.What are the roots of x² − 11x + 28 = 0? (a) 1 and 28 (b) 1 and 10 (c) 4 and 7 (d) 2 and 14
  32. 32.If a + b = 12 and ab = 20, what is (a − b)²? (a) 144 (b) 64 (c) 8 (d) 104
  33. 33.If a + b = 19 and ab = 90, what is a² + b²? (a) 541 (b) 361 (c) 271 (d) 181
  34. 34.Which is the correct factorisation of x² + 7x + 12? (a) (x + 3)(x + 4) (b) (x − 3)(x − 4) (c) (x + 1)(x + 12) (d) (x + 1)(x + 6)
  35. 35.What are the roots of x² − 12x + 35 = 0? (a) 2 and 10 (b) 1 and 11 (c) 1 and 35 (d) 5 and 7
  36. 36.If a + b = 16 and ab = 55, what is a² + b²? (a) 146 (b) 201 (c) 366 (d) 256
  37. 37.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 884 (b) 676 (c) 1156 (d) 686
  38. 38.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 5929 (b) 6889 (c) 6899 (d) 6409
  39. 39.What is the product of the roots of x² − 8x + 12 = 0? (a) 12 (b) 8 (c) 13 (d) 11
  40. 40.If a + b = 10 and ab = 9, what is (a − b)²? (a) 8 (b) 100 (c) 64 (d) 82

Answer key

  1. 1
  2. 68
  3. 208
  4. 2 and 5
  5. 4
  6. 7225
  7. 125
  8. 1296
  9. (x + 8)(x + 9)
  10. (x + 4)(x + 5)
  11. 4761
  12. (x + 6)(x + 7)
  13. 25
  14. 1225
  15. (x + 3)(x + 6)
  16. 54
  17. 16
  18. 10
  19. 1369
  20. 25
  21. 289
  22. (x + 2)(x + 4)
  23. (x + 4)(x + 8)
  24. 20
  25. 26
  26. 2916
  27. 20
  28. 36
  29. 40
  30. 13
  31. 4 and 7
  32. 64
  33. 181
  34. (x + 3)(x + 4)
  35. 5 and 7
  36. 146
  37. 676
  38. 6889
  39. 12
  40. 64

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