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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 209 · 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 = 4 and ab = 3, what is a² + b²? (a) 22 (b) 16 (c) 13 (d) 10
  2. 2.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x − 7)(x − 9) (c) (x + 1)(x + 15) (d) (x + 7)(x + 9)
  3. 3.If a + b = 20 and ab = 99, what is a² + b²? (a) 598 (b) 400 (c) 301 (d) 202
  4. 4.If a + b = 13 and ab = 42, what is (a − b)²? (a) 85 (b) 127 (c) 1 (d) 169
  5. 5.Which is the correct factorisation of x² + 10x + 21? (a) (x + 3)(x + 7) (b) (x − 3)(x − 7) (c) (x + 1)(x + 9) (d) (x + 1)(x + 21)
  6. 6.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 24) (b) (x + 1)(x + 9) (c) (x + 4)(x + 6) (d) (x − 4)(x − 6)
  7. 7.If a + b = 7 and ab = 10, what is (a − b)²? (a) 29 (b) 9 (c) 3 (d) 49
  8. 8.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 3616 (b) 4106 (c) 3136 (d) 4096
  9. 9.If a + b = 9 and ab = 20, what is (a − b)²? (a) 81 (b) 1 (c) 61 (d) 41
  10. 10.If a + b = 21 and ab = 108, what is (a − b)²? (a) 9 (b) 441 (c) 225 (d) 3
  11. 11.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6401 (b) 6241 (c) 6561 (d) 6571
  12. 12.Which is the correct factorisation of x² + 8x + 12? (a) (x − 2)(x − 6) (b) (x + 2)(x + 6) (c) (x + 1)(x + 12) (d) (x + 1)(x + 7)
  13. 13.If a + b = 16 and ab = 55, what is a² + b²? (a) 256 (b) 366 (c) 201 (d) 146
  14. 14.What is the sum of the roots of x² − 10x + 21 = 0? (a) 10 (b) 21 (c) 11 (d) 9
  15. 15.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 45) (b) (x − 5)(x − 9) (c) (x + 5)(x + 9) (d) (x + 1)(x + 13)
  16. 16.If a + b = 11 and ab = 30, what is a² + b²? (a) 121 (b) 181 (c) 61 (d) 91
  17. 17.What are the roots of x² − 9x + 18 = 0? (a) 2 and 9 (b) 3 and 6 (c) 1 and 18 (d) 1 and 8
  18. 18.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 10201 (b) 9999 (c) 9801 (d) 9811
  19. 19.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 9604 (b) 10004 (c) 10414 (d) 10404
  20. 20.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1521 (b) 1599 (c) 1681 (d) 1531
  21. 21.If a + b = 19 and ab = 84, what is a² + b²? (a) 193 (b) 277 (c) 529 (d) 361
  22. 22.If a + b = 16 and ab = 60, what is a² + b²? (a) 256 (b) 376 (c) 196 (d) 136
  23. 23.If a + b = 9 and ab = 8, what is (a − b)²? (a) 7 (b) 81 (c) 65 (d) 49
  24. 24.If a + b = 19 and ab = 90, what is (a − b)²? (a) 361 (b) 271 (c) 181 (d) 1
  25. 25.If a + b = 19 and ab = 88, what is (a − b)²? (a) 361 (b) 3 (c) 185 (d) 9
  26. 26.If a + b = 15 and ab = 44, what is (a − b)²? (a) 225 (b) 7 (c) 137 (d) 49
  27. 27.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 375 (b) 225 (c) 235 (d) 625
  28. 28.If a + b = 21 and ab = 108, what is a² + b²? (a) 333 (b) 225 (c) 441 (d) 657
  29. 29.If a + b = 3 and ab = 2, what is a² + b²? (a) 9 (b) 5 (c) 13 (d) 7
  30. 30.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4884 (b) 4356 (c) 5476 (d) 4366
  31. 31.Which is the correct factorisation of x² + 12x + 32? (a) (x − 4)(x − 8) (b) (x + 4)(x + 8) (c) (x + 1)(x + 32) (d) (x + 1)(x + 11)
  32. 32.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 1)(x + 42) (c) (x + 6)(x + 7) (d) (x + 1)(x + 12)
  33. 33.Which is the correct factorisation of x² + 7x + 10? (a) (x + 2)(x + 5) (b) (x − 2)(x − 5) (c) (x + 1)(x + 6) (d) (x + 1)(x + 10)
  34. 34.If a + b = 16 and ab = 63, what is (a − b)²? (a) 2 (b) 130 (c) 4 (d) 256
  35. 35.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)
  36. 36.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 8084 (b) 7406 (c) 7396 (d) 8836
  37. 37.If a + b = 10 and ab = 9, what is a² + b²? (a) 118 (b) 82 (c) 91 (d) 100
  38. 38.What are the roots of x² − 12x + 32 = 0? (a) 4 and 8 (b) 1 and 11 (c) 2 and 16 (d) 1 and 32
  39. 39.If a + b = 13 and ab = 22, what is (a − b)²? (a) 169 (b) 81 (c) 9 (d) 125
  40. 40.Which is the correct factorisation of x² + 10x + 16? (a) (x + 2)(x + 8) (b) (x + 1)(x + 16) (c) (x + 1)(x + 9) (d) (x − 2)(x − 8)

Answer key

  1. 10
  2. (x + 7)(x + 9)
  3. 202
  4. 1
  5. (x + 3)(x + 7)
  6. (x + 4)(x + 6)
  7. 9
  8. 4096
  9. 1
  10. 9
  11. 6561
  12. (x + 2)(x + 6)
  13. 146
  14. 10
  15. (x + 5)(x + 9)
  16. 61
  17. 3 and 6
  18. 9801
  19. 10404
  20. 1521
  21. 193
  22. 136
  23. 49
  24. 1
  25. 9
  26. 49
  27. 225
  28. 225
  29. 5
  30. 4356
  31. (x + 4)(x + 8)
  32. (x + 6)(x + 7)
  33. (x + 2)(x + 5)
  34. 4
  35. (x + 8)(x + 9)
  36. 7396
  37. 82
  38. 4 and 8
  39. 81
  40. (x + 2)(x + 8)

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