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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 847 · 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.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 8836 (b) 8116 (c) 8846 (d) 7396
  2. 2.If a + b = 8 and ab = 12, what is a² + b²? (a) 64 (b) 52 (c) 88 (d) 40
  3. 3.What is the sum of the roots of x² − 15x + 54 = 0? (a) 14 (b) 15 (c) 16 (d) 54
  4. 4.Which is the correct factorisation of x² + 9x + 18? (a) (x + 3)(x + 6) (b) (x + 1)(x + 18) (c) (x + 1)(x + 8) (d) (x − 3)(x − 6)
  5. 5.What is the product of the roots of x² − 12x + 32 = 0? (a) 32 (b) 33 (c) 12 (d) 31
  6. 6.What are the roots of x² − 6x + 8 = 0? (a) 1 and 8 (b) 1 and 5 (c) 2 and 5 (d) 2 and 4
  7. 7.If a + b = 13 and ab = 36, what is a² + b²? (a) 133 (b) 169 (c) 97 (d) 241
  8. 8.If a + b = 8 and ab = 15, what is (a − b)²? (a) 2 (b) 34 (c) 4 (d) 64
  9. 9.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2496 (b) 2304 (c) 2314 (d) 2704
  10. 10.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x − 5)(x − 7) (c) (x + 5)(x + 7) (d) (x + 1)(x + 35)
  11. 11.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 3625 (b) 4225 (c) 3025 (d) 4235
  12. 12.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8104 (b) 7744 (c) 8474 (d) 8464
  13. 13.If a + b = 19 and ab = 90, what is (a − b)²? (a) 181 (b) 361 (c) 271 (d) 1
  14. 14.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 2525 (b) 3035 (c) 2025 (d) 3025
  15. 15.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 576 (b) 586 (c) 416 (d) 256
  16. 16.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 10016 (b) 10826 (c) 10816 (d) 9216
  17. 17.If a + b = 10 and ab = 9, what is (a − b)²? (a) 82 (b) 64 (c) 8 (d) 100
  18. 18.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 4489 (b) 4909 (c) 5329 (d) 5339
  19. 19.If a + b = 5 and ab = 6, what is a² + b²? (a) 25 (b) 37 (c) 19 (d) 13
  20. 20.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3721 (b) 3601 (c) 3731 (d) 3481
  21. 21.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 404 (b) 494 (c) 484 (d) 324
  22. 22.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x + 1)(x + 15) (c) (x + 1)(x + 63) (d) (x − 7)(x − 9)
  23. 23.If a + b = 19 and ab = 84, what is (a − b)²? (a) 361 (b) 193 (c) 25 (d) 5
  24. 24.If a + b = 12 and ab = 20, what is (a − b)²? (a) 8 (b) 104 (c) 144 (d) 64
  25. 25.Which is the correct factorisation of x² + 13x + 36? (a) (x − 4)(x − 9) (b) (x + 1)(x + 36) (c) (x + 1)(x + 12) (d) (x + 4)(x + 9)
  26. 26.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 1575 (b) 2025 (c) 1225 (d) 1235
  27. 27.If a + b = 16 and ab = 60, what is (a − b)²? (a) 136 (b) 256 (c) 4 (d) 16
  28. 28.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)
  29. 29.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)
  30. 30.What is the product of the roots of x² − 9x + 14 = 0? (a) 9 (b) 15 (c) 13 (d) 14
  31. 31.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 21) (b) (x − 3)(x − 7) (c) (x + 3)(x + 7) (d) (x + 1)(x + 9)
  32. 32.If a + b = 16 and ab = 63, what is a² + b²? (a) 130 (b) 256 (c) 382 (d) 193
  33. 33.If a + b = 16 and ab = 55, what is a² + b²? (a) 201 (b) 256 (c) 146 (d) 366
  34. 34.What is the sum of the roots of x² − 14x + 45 = 0? (a) 15 (b) 14 (c) 45 (d) 13
  35. 35.If a + b = 17 and ab = 66, what is a² + b²? (a) 223 (b) 157 (c) 289 (d) 421
  36. 36.If a + b = 11 and ab = 30, what is a² + b²? (a) 181 (b) 121 (c) 91 (d) 61
  37. 37.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1296 (b) 1946 (c) 1616 (d) 1936
  38. 38.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 17 (c) 16 (d) 72
  39. 39.If a + b = 4 and ab = 3, what is a² + b²? (a) 22 (b) 13 (c) 16 (d) 10
  40. 40.What is the sum of the roots of x² − 15x + 56 = 0? (a) 16 (b) 14 (c) 56 (d) 15

Answer key

  1. 8836
  2. 40
  3. 15
  4. (x + 3)(x + 6)
  5. 32
  6. 2 and 4
  7. 97
  8. 4
  9. 2304
  10. (x + 5)(x + 7)
  11. 4225
  12. 8464
  13. 1
  14. 3025
  15. 576
  16. 10816
  17. 64
  18. 5329
  19. 13
  20. 3721
  21. 484
  22. (x + 7)(x + 9)
  23. 25
  24. 64
  25. (x + 4)(x + 9)
  26. 1225
  27. 16
  28. (x + 4)(x + 8)
  29. (x + 7)(x + 8)
  30. 14
  31. (x + 3)(x + 7)
  32. 130
  33. 146
  34. 14
  35. 157
  36. 61
  37. 1936
  38. 17
  39. 10
  40. 15

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