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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 947 · 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 82²? (a) 6724 (b) 6084 (c) 6404 (d) 6734
  2. 2.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 6416 (b) 5776 (c) 7066 (d) 7056
  3. 3.What is the product of the roots of x² − 13x + 42 = 0? (a) 13 (b) 43 (c) 42 (d) 41
  4. 4.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 13) (b) (x − 5)(x − 9) (c) (x + 1)(x + 45) (d) (x + 5)(x + 9)
  5. 5.Which is the correct factorisation of x² + 13x + 36? (a) (x + 4)(x + 9) (b) (x + 1)(x + 12) (c) (x + 1)(x + 36) (d) (x − 4)(x − 9)
  6. 6.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 1166 (b) 916 (c) 1156 (d) 676
  7. 7.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x + 1)(x + 56) (c) (x + 1)(x + 14) (d) (x − 7)(x − 8)
  8. 8.If a + b = 16 and ab = 60, what is a² + b²? (a) 376 (b) 196 (c) 256 (d) 136
  9. 9.If a + b = 18 and ab = 77, what is (a − b)²? (a) 16 (b) 4 (c) 324 (d) 170
  10. 10.What are the roots of x² − 9x + 18 = 0? (a) 2 and 9 (b) 1 and 8 (c) 3 and 6 (d) 1 and 18
  11. 11.What is the sum of the roots of x² − 13x + 42 = 0? (a) 14 (b) 13 (c) 42 (d) 12
  12. 12.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1454 (b) 1764 (c) 1596 (d) 1444
  13. 13.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 1)(x + 72) (c) (x − 8)(x − 9) (d) (x + 8)(x + 9)
  14. 14.If a + b = 19 and ab = 84, what is a² + b²? (a) 529 (b) 277 (c) 193 (d) 361
  15. 15.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 24) (b) (x − 3)(x − 8) (c) (x + 1)(x + 10) (d) (x + 3)(x + 8)
  16. 16.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 12) (b) (x − 2)(x − 6) (c) (x + 2)(x + 6) (d) (x + 1)(x + 7)
  17. 17.What is the product of the roots of x² − 14x + 48 = 0? (a) 48 (b) 14 (c) 47 (d) 49
  18. 18.If a + b = 13 and ab = 36, what is (a − b)²? (a) 25 (b) 5 (c) 169 (d) 97
  19. 19.If a + b = 21 and ab = 108, what is a² + b²? (a) 657 (b) 333 (c) 441 (d) 225
  20. 20.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
  21. 21.If a + b = 14 and ab = 40, what is (a − b)²? (a) 116 (b) 6 (c) 36 (d) 196
  22. 22.What is the sum of the roots of x² − 16x + 63 = 0? (a) 15 (b) 16 (c) 17 (d) 63
  23. 23.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 7)(x + 9) (c) (x + 1)(x + 63) (d) (x − 7)(x − 9)
  24. 24.If a + b = 10 and ab = 24, what is a² + b²? (a) 148 (b) 100 (c) 76 (d) 52
  25. 25.If a + b = 16 and ab = 48, what is a² + b²? (a) 256 (b) 160 (c) 352 (d) 208
  26. 26.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 11) (b) (x − 4)(x − 8) (c) (x + 1)(x + 32) (d) (x + 4)(x + 8)
  27. 27.What is the sum of the roots of x² − 13x + 36 = 0? (a) 36 (b) 14 (c) 13 (d) 12
  28. 28.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 8099 (b) 7931 (c) 7921 (d) 8281
  29. 29.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 729 (b) 1089 (c) 909 (d) 1099
  30. 30.If a + b = 12 and ab = 32, what is (a − b)²? (a) 80 (b) 4 (c) 144 (d) 16
  31. 31.If a + b = 20 and ab = 96, what is a² + b²? (a) 400 (b) 208 (c) 592 (d) 304
  32. 32.What is the sum of the roots of x² − 11x + 28 = 0? (a) 10 (b) 11 (c) 12 (d) 28
  33. 33.What are the roots of x² − 6x + 8 = 0? (a) 1 and 5 (b) 2 and 4 (c) 2 and 5 (d) 1 and 8
  34. 34.If a + b = 19 and ab = 90, what is a² + b²? (a) 361 (b) 181 (c) 541 (d) 271
  35. 35.If a + b = 19 and ab = 88, what is (a − b)²? (a) 185 (b) 361 (c) 9 (d) 3
  36. 36.If a + b = 11 and ab = 28, what is (a − b)²? (a) 9 (b) 3 (c) 65 (d) 121
  37. 37.If a + b = 8 and ab = 15, what is a² + b²? (a) 64 (b) 34 (c) 49 (d) 94
  38. 38.What are the roots of x² − 14x + 48 = 0? (a) 6 and 8 (b) 2 and 24 (c) 1 and 48 (d) 1 and 13
  39. 39.What is the sum of the roots of x² − 7x + 12 = 0? (a) 8 (b) 12 (c) 6 (d) 7
  40. 40.If a + b = 17 and ab = 60, what is a² + b²? (a) 229 (b) 169 (c) 409 (d) 289

Answer key

  1. 6724
  2. 7056
  3. 42
  4. (x + 5)(x + 9)
  5. (x + 4)(x + 9)
  6. 1156
  7. (x + 7)(x + 8)
  8. 136
  9. 16
  10. 3 and 6
  11. 13
  12. 1444
  13. (x + 8)(x + 9)
  14. 193
  15. (x + 3)(x + 8)
  16. (x + 2)(x + 6)
  17. 48
  18. 25
  19. 225
  20. 5 and 7
  21. 36
  22. 16
  23. (x + 7)(x + 9)
  24. 52
  25. 160
  26. (x + 4)(x + 8)
  27. 13
  28. 7921
  29. 1089
  30. 16
  31. 208
  32. 11
  33. 2 and 4
  34. 181
  35. 9
  36. 9
  37. 34
  38. 6 and 8
  39. 7
  40. 169

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