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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 611 · 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.What are the roots of x² − 17x + 72 = 0? (a) 8 and 9 (b) 1 and 16 (c) 1 and 72 (d) 2 and 36
  2. 2.What is the sum of the roots of x² − 14x + 48 = 0? (a) 48 (b) 15 (c) 14 (d) 13
  3. 3.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6396 (b) 6724 (c) 6084 (d) 6094
  4. 4.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x − 8)(x − 9) (c) (x + 1)(x + 16) (d) (x + 1)(x + 72)
  5. 5.If a + b = 14 and ab = 48, what is (a − b)²? (a) 2 (b) 100 (c) 196 (d) 4
  6. 6.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 6409 (b) 6889 (c) 6899 (d) 5929
  7. 7.What are the roots of x² − 13x + 42 = 0? (a) 1 and 12 (b) 6 and 7 (c) 2 and 21 (d) 1 and 42
  8. 8.What are the roots of x² − 15x + 56 = 0? (a) 2 and 28 (b) 1 and 14 (c) 1 and 56 (d) 7 and 8
  9. 9.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)
  10. 10.What is the sum of the roots of x² − 16x + 63 = 0? (a) 17 (b) 63 (c) 16 (d) 15
  11. 11.If a + b = 19 and ab = 90, what is a² + b²? (a) 361 (b) 541 (c) 271 (d) 181
  12. 12.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 10) (b) (x + 2)(x + 5) (c) (x + 1)(x + 6) (d) (x − 2)(x − 5)
  13. 13.What is the product of the roots of x² − 15x + 56 = 0? (a) 55 (b) 56 (c) 15 (d) 57
  14. 14.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6561 (b) 6251 (c) 6399 (d) 6241
  15. 15.If a + b = 11 and ab = 24, what is a² + b²? (a) 73 (b) 97 (c) 169 (d) 121
  16. 16.If a + b = 12 and ab = 32, what is (a − b)²? (a) 16 (b) 144 (c) 4 (d) 80
  17. 17.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x − 7)(x − 9) (c) (x + 1)(x + 63) (d) (x + 1)(x + 15)
  18. 18.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 71 (c) 73 (d) 72
  19. 19.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 7) (b) (x − 2)(x − 6) (c) (x + 2)(x + 6) (d) (x + 1)(x + 12)
  20. 20.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 7225 (b) 6425 (c) 7235 (d) 5625
  21. 21.If a + b = 15 and ab = 44, what is a² + b²? (a) 137 (b) 313 (c) 181 (d) 225
  22. 22.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10004 (b) 10414 (c) 10404 (d) 9604
  23. 23.If a + b = 10 and ab = 16, what is (a − b)²? (a) 100 (b) 68 (c) 6 (d) 36
  24. 24.If a + b = 10 and ab = 21, what is (a − b)²? (a) 100 (b) 16 (c) 58 (d) 4
  25. 25.If a + b = 12 and ab = 27, what is a² + b²? (a) 90 (b) 198 (c) 117 (d) 144
  26. 26.If a + b = 13 and ab = 30, what is (a − b)²? (a) 109 (b) 169 (c) 7 (d) 49
  27. 27.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 1156 (b) 916 (c) 1166 (d) 676
  28. 28.If a + b = 16 and ab = 63, what is (a − b)²? (a) 130 (b) 256 (c) 2 (d) 4
  29. 29.Which is the correct factorisation of x² + 13x + 42? (a) (x + 6)(x + 7) (b) (x + 1)(x + 12) (c) (x + 1)(x + 42) (d) (x − 6)(x − 7)
  30. 30.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3249 (b) 3259 (c) 3591 (d) 3969
  31. 31.Which is the correct factorisation of x² + 15x + 54? (a) (x + 6)(x + 9) (b) (x + 1)(x + 14) (c) (x + 1)(x + 54) (d) (x − 6)(x − 9)
  32. 32.What is the product of the roots of x² − 8x + 12 = 0? (a) 11 (b) 12 (c) 13 (d) 8
  33. 33.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8281 (b) 8101 (c) 7921 (d) 8291
  34. 34.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 451 (b) 441 (c) 401 (d) 361
  35. 35.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 404 (b) 494 (c) 484 (d) 324
  36. 36.If a + b = 13 and ab = 40, what is a² + b²? (a) 169 (b) 249 (c) 129 (d) 89
  37. 37.If a + b = 6 and ab = 8, what is (a − b)²? (a) 36 (b) 2 (c) 20 (d) 4
  38. 38.If a + b = 19 and ab = 88, what is a² + b²? (a) 537 (b) 361 (c) 273 (d) 185
  39. 39.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 17 (c) 16 (d) 18
  40. 40.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 9216 (b) 9226 (c) 10816 (d) 9984

Answer key

  1. 8 and 9
  2. 14
  3. 6084
  4. (x + 8)(x + 9)
  5. 4
  6. 6889
  7. 6 and 7
  8. 7 and 8
  9. (x + 7)(x + 8)
  10. 16
  11. 181
  12. (x + 2)(x + 5)
  13. 56
  14. 6241
  15. 73
  16. 16
  17. (x + 7)(x + 9)
  18. 72
  19. (x + 2)(x + 6)
  20. 7225
  21. 137
  22. 10404
  23. 36
  24. 16
  25. 90
  26. 49
  27. 1156
  28. 4
  29. (x + 6)(x + 7)
  30. 3249
  31. (x + 6)(x + 9)
  32. 12
  33. 8281
  34. 441
  35. 484
  36. 89
  37. 4
  38. 185
  39. 17
  40. 9216

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