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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 536 · 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 + 5x + 6? (a) (x + 1)(x + 4) (b) (x − 2)(x − 3) (c) (x + 2)(x + 3) (d) (x + 1)(x + 6)
  2. 2.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x + 1)(x + 72) (c) (x + 1)(x + 16) (d) (x − 8)(x − 9)
  3. 3.Which is the correct factorisation of x² + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 1)(x + 6) (c) (x + 3)(x + 4) (d) (x + 1)(x + 12)
  4. 4.Which is the correct factorisation of x² + 6x + 8? (a) (x − 2)(x − 4) (b) (x + 2)(x + 4) (c) (x + 1)(x + 8) (d) (x + 1)(x + 5)
  5. 5.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2126 (b) 2484 (c) 2916 (d) 2116
  6. 6.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 10) (b) (x + 3)(x + 8) (c) (x − 3)(x − 8) (d) (x + 1)(x + 24)
  7. 7.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3979 (b) 3969 (c) 3609 (d) 3249
  8. 8.What is the sum of the roots of x² − 12x + 32 = 0? (a) 11 (b) 32 (c) 13 (d) 12
  9. 9.What are the roots of x² − 5x + 6 = 0? (a) 2 and 4 (b) 1 and 6 (c) 1 and 4 (d) 2 and 3
  10. 10.If a + b = 21 and ab = 108, what is (a − b)²? (a) 441 (b) 9 (c) 3 (d) 225
  11. 11.What is the sum of the roots of x² − 10x + 16 = 0? (a) 11 (b) 16 (c) 10 (d) 9
  12. 12.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 6416 (b) 7056 (c) 7066 (d) 5776
  13. 13.If a + b = 14 and ab = 33, what is a² + b²? (a) 262 (b) 163 (c) 196 (d) 130
  14. 14.If a + b = 11 and ab = 30, what is (a − b)²? (a) 91 (b) 61 (c) 1 (d) 121
  15. 15.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 63) (c) (x + 1)(x + 15) (d) (x + 7)(x + 9)
  16. 16.If a + b = 5 and ab = 6, what is a² + b²? (a) 37 (b) 25 (c) 19 (d) 13
  17. 17.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x − 5)(x − 7) (c) (x + 1)(x + 35) (d) (x + 5)(x + 7)
  18. 18.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)
  19. 19.If a + b = 7 and ab = 12, what is (a − b)²? (a) 49 (b) 37 (c) 1 (d) 25
  20. 20.What is the sum of the roots of x² − 10x + 24 = 0? (a) 10 (b) 9 (c) 24 (d) 11
  21. 21.If a + b = 5 and ab = 4, what is a² + b²? (a) 33 (b) 25 (c) 17 (d) 21
  22. 22.If a + b = 12 and ab = 27, what is (a − b)²? (a) 90 (b) 6 (c) 36 (d) 144
  23. 23.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 71 (c) 72 (d) 17
  24. 24.If a + b = 15 and ab = 56, what is (a − b)²? (a) 113 (b) 169 (c) 225 (d) 1
  25. 25.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1599 (b) 1521 (c) 1681 (d) 1531
  26. 26.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1691 (b) 1601 (c) 1681 (d) 1521
  27. 27.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 916 (b) 1166 (c) 1156 (d) 676
  28. 28.If a + b = 20 and ab = 99, what is (a − b)²? (a) 2 (b) 4 (c) 202 (d) 400
  29. 29.If a + b = 15 and ab = 54, what is a² + b²? (a) 171 (b) 225 (c) 117 (d) 333
  30. 30.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x + 1)(x + 12) (c) (x + 6)(x + 7) (d) (x − 6)(x − 7)
  31. 31.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)
  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 + 13) (d) (x + 1)(x + 48)
  33. 33.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 9991 (b) 9409 (c) 9419 (d) 10609
  34. 34.If a + b = 15 and ab = 36, what is (a − b)²? (a) 81 (b) 9 (c) 153 (d) 225
  35. 35.If a + b = 16 and ab = 48, what is a² + b²? (a) 256 (b) 208 (c) 160 (d) 352
  36. 36.If a + b = 18 and ab = 77, what is a² + b²? (a) 478 (b) 247 (c) 170 (d) 324
  37. 37.If a + b = 19 and ab = 88, what is a² + b²? (a) 361 (b) 273 (c) 537 (d) 185
  38. 38.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 925 (b) 1225 (c) 625 (d) 1235
  39. 39.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 7396 (b) 8836 (c) 7406 (d) 8084
  40. 40.Which is the correct factorisation of x² + 13x + 40? (a) (x − 5)(x − 8) (b) (x + 1)(x + 40) (c) (x + 5)(x + 8) (d) (x + 1)(x + 12)

Answer key

  1. (x + 2)(x + 3)
  2. (x + 8)(x + 9)
  3. (x + 3)(x + 4)
  4. (x + 2)(x + 4)
  5. 2116
  6. (x + 3)(x + 8)
  7. 3969
  8. 12
  9. 2 and 3
  10. 9
  11. 10
  12. 7056
  13. 130
  14. 1
  15. (x + 7)(x + 9)
  16. 13
  17. (x + 5)(x + 7)
  18. (x + 4)(x + 5)
  19. 1
  20. 10
  21. 17
  22. 36
  23. 72
  24. 1
  25. 1521
  26. 1681
  27. 1156
  28. 4
  29. 117
  30. (x + 6)(x + 7)
  31. (x + 4)(x + 8)
  32. (x + 6)(x + 8)
  33. 9409
  34. 81
  35. 160
  36. 170
  37. 185
  38. 1225
  39. 7396
  40. (x + 5)(x + 8)

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