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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 753 · 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 = 11 and ab = 24, what is a² + b²? (a) 121 (b) 97 (c) 169 (d) 73
  2. 2.If a + b = 17 and ab = 66, what is a² + b²? (a) 223 (b) 289 (c) 157 (d) 421
  3. 3.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 11) (b) (x + 3)(x + 9) (c) (x − 3)(x − 9) (d) (x + 1)(x + 27)
  4. 4.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 9801 (b) 9811 (c) 9999 (d) 10201
  5. 5.What is the sum of the roots of x² − 13x + 40 = 0? (a) 14 (b) 12 (c) 13 (d) 40
  6. 6.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6251 (b) 6561 (c) 6399 (d) 6241
  7. 7.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1849 (b) 1369 (c) 1379 (d) 1591
  8. 8.If a + b = 19 and ab = 84, what is a² + b²? (a) 361 (b) 529 (c) 277 (d) 193
  9. 9.If a + b = 11 and ab = 30, what is (a − b)²? (a) 61 (b) 91 (c) 121 (d) 1
  10. 10.If a + b = 16 and ab = 55, what is a² + b²? (a) 256 (b) 201 (c) 366 (d) 146
  11. 11.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 3136 (b) 4096 (c) 3616 (d) 4106
  12. 12.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 1156 (b) 676 (c) 916 (d) 1166
  13. 13.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 7225 (b) 6425 (c) 5625 (d) 7235
  14. 14.What is the product of the roots of x² − 5x + 6 = 0? (a) 8 (b) 7 (c) 5 (d) 6
  15. 15.If a + b = 11 and ab = 30, what is a² + b²? (a) 121 (b) 181 (c) 91 (d) 61
  16. 16.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 1225 (b) 875 (c) 635 (d) 625
  17. 17.What is the sum of the roots of x² − 13x + 42 = 0? (a) 42 (b) 13 (c) 12 (d) 14
  18. 18.What are the roots of x² − 12x + 27 = 0? (a) 1 and 27 (b) 2 and 10 (c) 3 and 9 (d) 1 and 11
  19. 19.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 8) (b) (x + 1)(x + 5) (c) (x − 2)(x − 4) (d) (x + 2)(x + 4)
  20. 20.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x + 1)(x + 8) (c) (x + 4)(x + 5) (d) (x − 4)(x − 5)
  21. 21.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 10) (b) (x − 4)(x − 7) (c) (x + 1)(x + 28) (d) (x + 4)(x + 7)
  22. 22.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 1)(x + 16) (c) (x + 8)(x + 9) (d) (x − 8)(x − 9)
  23. 23.If a + b = 20 and ab = 99, what is (a − b)²? (a) 2 (b) 4 (c) 202 (d) 400
  24. 24.What is the sum of the roots of x² − 8x + 12 = 0? (a) 12 (b) 9 (c) 8 (d) 7
  25. 25.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 18 (c) 17 (d) 16
  26. 26.If a + b = 16 and ab = 60, what is (a − b)²? (a) 256 (b) 136 (c) 4 (d) 16
  27. 27.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 12) (b) (x − 2)(x − 6) (c) (x + 1)(x + 7) (d) (x + 2)(x + 6)
  28. 28.If a + b = 3 and ab = 2, what is (a − b)²? (a) 5 (b) 9 (c) 7 (d) 1
  29. 29.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 6391 (b) 5929 (c) 5939 (d) 6889
  30. 30.What are the roots of x² − 11x + 30 = 0? (a) 5 and 6 (b) 1 and 10 (c) 2 and 15 (d) 1 and 30
  31. 31.What are the roots of x² − 10x + 21 = 0? (a) 1 and 21 (b) 1 and 9 (c) 3 and 7 (d) 2 and 8
  32. 32.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10201 (b) 9801 (c) 10001 (d) 10211
  33. 33.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x + 1)(x + 35) (c) (x − 5)(x − 7) (d) (x + 5)(x + 7)
  34. 34.If a + b = 18 and ab = 72, what is a² + b²? (a) 468 (b) 180 (c) 324 (d) 252
  35. 35.If a + b = 12 and ab = 27, what is (a − b)²? (a) 36 (b) 90 (c) 6 (d) 144
  36. 36.What is the sum of the roots of x² − 12x + 35 = 0? (a) 35 (b) 12 (c) 13 (d) 11
  37. 37.What is the product of the roots of x² − 17x + 72 = 0? (a) 72 (b) 71 (c) 17 (d) 73
  38. 38.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 539 (b) 289 (c) 529 (d) 409
  39. 39.If a + b = 14 and ab = 48, what is a² + b²? (a) 196 (b) 292 (c) 148 (d) 100
  40. 40.If a + b = 7 and ab = 10, what is a² + b²? (a) 29 (b) 69 (c) 39 (d) 49

Answer key

  1. 73
  2. 157
  3. (x + 3)(x + 9)
  4. 9801
  5. 13
  6. 6241
  7. 1369
  8. 193
  9. 1
  10. 146
  11. 4096
  12. 1156
  13. 7225
  14. 6
  15. 61
  16. 625
  17. 13
  18. 3 and 9
  19. (x + 2)(x + 4)
  20. (x + 4)(x + 5)
  21. (x + 4)(x + 7)
  22. (x + 8)(x + 9)
  23. 4
  24. 8
  25. 17
  26. 16
  27. (x + 2)(x + 6)
  28. 1
  29. 5929
  30. 5 and 6
  31. 3 and 7
  32. 10201
  33. (x + 5)(x + 7)
  34. 180
  35. 36
  36. 12
  37. 72
  38. 529
  39. 100
  40. 29

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