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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 958 · 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 is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 17 (c) 72 (d) 16
  2. 2.If a + b = 12 and ab = 35, what is (a − b)²? (a) 2 (b) 4 (c) 74 (d) 144
  3. 3.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 7056 (b) 5776 (c) 5786 (d) 6384
  4. 4.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)
  5. 5.What is the sum of the roots of x² − 13x + 36 = 0? (a) 13 (b) 14 (c) 36 (d) 12
  6. 6.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1306 (b) 1584 (c) 1936 (d) 1296
  7. 7.If a + b = 18 and ab = 77, what is (a − b)²? (a) 4 (b) 170 (c) 324 (d) 16
  8. 8.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3259 (b) 3969 (c) 3591 (d) 3249
  9. 9.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3364 (b) 3854 (c) 3604 (d) 3844
  10. 10.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10004 (b) 10404 (c) 10414 (d) 9604
  11. 11.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6561 (b) 6241 (c) 6399 (d) 6251
  12. 12.If a + b = 12 and ab = 32, what is (a − b)²? (a) 16 (b) 144 (c) 80 (d) 4
  13. 13.If a + b = 15 and ab = 56, what is (a − b)²? (a) 1 (b) 225 (c) 169 (d) 113
  14. 14.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)
  15. 15.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 899 (b) 961 (c) 841 (d) 851
  16. 16.If a + b = 8 and ab = 15, what is a² + b²? (a) 94 (b) 49 (c) 64 (d) 34
  17. 17.Which is the correct factorisation of x² + 9x + 18? (a) (x − 3)(x − 6) (b) (x + 1)(x + 18) (c) (x + 3)(x + 6) (d) (x + 1)(x + 8)
  18. 18.If a + b = 19 and ab = 90, what is a² + b²? (a) 181 (b) 361 (c) 541 (d) 271
  19. 19.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 17 (c) 72 (d) 73
  20. 20.If a + b = 11 and ab = 28, what is a² + b²? (a) 121 (b) 93 (c) 177 (d) 65
  21. 21.If a + b = 18 and ab = 77, what is a² + b²? (a) 170 (b) 478 (c) 247 (d) 324
  22. 22.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x − 5)(x − 6) (c) (x + 1)(x + 30) (d) (x + 5)(x + 6)
  23. 23.What is the sum of the roots of x² − 10x + 21 = 0? (a) 21 (b) 10 (c) 11 (d) 9
  24. 24.If a + b = 12 and ab = 35, what is a² + b²? (a) 109 (b) 144 (c) 214 (d) 74
  25. 25.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x + 1)(x + 15) (c) (x + 7)(x + 9) (d) (x − 7)(x − 9)
  26. 26.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 8)(x + 9) (c) (x + 1)(x + 72) (d) (x + 1)(x + 16)
  27. 27.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 8281 (b) 7921 (c) 8099 (d) 7931
  28. 28.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 4489 (b) 5329 (c) 4909 (d) 5339
  29. 29.What is the sum of the roots of x² − 11x + 30 = 0? (a) 12 (b) 30 (c) 10 (d) 11
  30. 30.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8101 (b) 8281 (c) 8291 (d) 7921
  31. 31.What is the sum of the roots of x² − 10x + 24 = 0? (a) 24 (b) 9 (c) 11 (d) 10
  32. 32.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 35) (b) (x + 1)(x + 11) (c) (x − 5)(x − 7) (d) (x + 5)(x + 7)
  33. 33.What is the product of the roots of x² − 7x + 12 = 0? (a) 11 (b) 12 (c) 13 (d) 7
  34. 34.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2314 (b) 2704 (c) 2304 (d) 2496
  35. 35.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3599 (b) 3491 (c) 3481 (d) 3721
  36. 36.What are the roots of x² − 11x + 28 = 0? (a) 1 and 28 (b) 2 and 14 (c) 1 and 10 (d) 4 and 7
  37. 37.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 36) (b) (x + 1)(x + 12) (c) (x − 4)(x − 9) (d) (x + 4)(x + 9)
  38. 38.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 1089 (b) 891 (c) 739 (d) 729
  39. 39.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6571 (b) 6241 (c) 6561 (d) 6401
  40. 40.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 5625 (b) 7225 (c) 7235 (d) 6425

Answer key

  1. 17
  2. 4
  3. 5776
  4. (x + 4)(x + 6)
  5. 13
  6. 1296
  7. 16
  8. 3249
  9. 3844
  10. 10404
  11. 6241
  12. 16
  13. 1
  14. (x + 3)(x + 8)
  15. 841
  16. 34
  17. (x + 3)(x + 6)
  18. 181
  19. 72
  20. 65
  21. 170
  22. (x + 5)(x + 6)
  23. 10
  24. 74
  25. (x + 7)(x + 9)
  26. (x + 8)(x + 9)
  27. 7921
  28. 5329
  29. 11
  30. 8281
  31. 10
  32. (x + 5)(x + 7)
  33. 12
  34. 2304
  35. 3481
  36. 4 and 7
  37. (x + 4)(x + 9)
  38. 729
  39. 6561
  40. 7225

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