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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 951 · 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 25²? (a) 635 (b) 625 (c) 225 (d) 425
  2. 2.If a + b = 20 and ab = 96, what is (a − b)²? (a) 400 (b) 4 (c) 16 (d) 208
  3. 3.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 1089 (b) 891 (c) 739 (d) 729
  4. 4.Which is the correct factorisation of x² + 9x + 20? (a) (x − 4)(x − 5) (b) (x + 1)(x + 20) (c) (x + 1)(x + 8) (d) (x + 4)(x + 5)
  5. 5.If a + b = 18 and ab = 72, what is (a − b)²? (a) 180 (b) 6 (c) 324 (d) 36
  6. 6.If a + b = 17 and ab = 72, what is a² + b²? (a) 289 (b) 433 (c) 217 (d) 145
  7. 7.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3609 (b) 3249 (c) 3969 (d) 3979
  8. 8.What is the product of the roots of x² − 7x + 10 = 0? (a) 7 (b) 10 (c) 11 (d) 9
  9. 9.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2304 (b) 2704 (c) 2714 (d) 2504
  10. 10.If a + b = 20 and ab = 99, what is a² + b²? (a) 301 (b) 202 (c) 598 (d) 400
  11. 11.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 8125 (b) 7225 (c) 9025 (d) 9035
  12. 12.If a + b = 17 and ab = 72, what is (a − b)²? (a) 217 (b) 145 (c) 289 (d) 1
  13. 13.If a + b = 13 and ab = 36, what is (a − b)²? (a) 169 (b) 25 (c) 97 (d) 5
  14. 14.If a + b = 9 and ab = 18, what is a² + b²? (a) 81 (b) 45 (c) 117 (d) 63
  15. 15.If a + b = 17 and ab = 70, what is (a − b)²? (a) 9 (b) 289 (c) 149 (d) 3
  16. 16.If a + b = 13 and ab = 30, what is a² + b²? (a) 229 (b) 139 (c) 109 (d) 169
  17. 17.If a + b = 21 and ab = 108, what is a² + b²? (a) 441 (b) 333 (c) 225 (d) 657
  18. 18.What is the sum of the roots of x² − 16x + 63 = 0? (a) 15 (b) 17 (c) 16 (d) 63
  19. 19.What is the sum of the roots of x² − 15x + 54 = 0? (a) 15 (b) 16 (c) 14 (d) 54
  20. 20.If a + b = 6 and ab = 5, what is a² + b²? (a) 36 (b) 31 (c) 26 (d) 46
  21. 21.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 8096 (b) 7754 (c) 7744 (d) 8464
  22. 22.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3374 (b) 3844 (c) 3596 (d) 3364
  23. 23.What are the roots of x² − 11x + 30 = 0? (a) 2 and 15 (b) 1 and 30 (c) 5 and 6 (d) 1 and 10
  24. 24.If a + b = 10 and ab = 9, what is (a − b)²? (a) 82 (b) 100 (c) 64 (d) 8
  25. 25.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6094 (b) 6724 (c) 6084 (d) 6396
  26. 26.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 289 (b) 539 (c) 529 (d) 409
  27. 27.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)
  28. 28.If a + b = 15 and ab = 44, what is a² + b²? (a) 313 (b) 181 (c) 137 (d) 225
  29. 29.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 4225 (b) 4925 (c) 5625 (d) 5635
  30. 30.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 4901 (b) 5041 (c) 5051 (d) 4761
  31. 31.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 451 (b) 361 (c) 441 (d) 401
  32. 32.If a + b = 19 and ab = 88, what is (a − b)²? (a) 9 (b) 361 (c) 185 (d) 3
  33. 33.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 5) (b) (x + 2)(x + 4) (c) (x − 2)(x − 4) (d) (x + 1)(x + 8)
  34. 34.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 896 (b) 794 (c) 1024 (d) 784
  35. 35.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3364 (b) 3604 (c) 3844 (d) 3854
  36. 36.What is the product of the roots of x² − 13x + 36 = 0? (a) 13 (b) 37 (c) 35 (d) 36
  37. 37.Which is the correct factorisation of x² + 11x + 28? (a) (x + 4)(x + 7) (b) (x + 1)(x + 28) (c) (x − 4)(x − 7) (d) (x + 1)(x + 10)
  38. 38.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 5776 (b) 7066 (c) 7056 (d) 6416
  39. 39.If a + b = 11 and ab = 10, what is a² + b²? (a) 141 (b) 121 (c) 111 (d) 101
  40. 40.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6251 (b) 6241 (c) 6399 (d) 6561

Answer key

  1. 625
  2. 16
  3. 729
  4. (x + 4)(x + 5)
  5. 36
  6. 145
  7. 3969
  8. 10
  9. 2704
  10. 202
  11. 9025
  12. 1
  13. 25
  14. 45
  15. 9
  16. 109
  17. 225
  18. 16
  19. 15
  20. 26
  21. 7744
  22. 3364
  23. 5 and 6
  24. 64
  25. 6084
  26. 529
  27. (x + 7)(x + 8)
  28. 137
  29. 5625
  30. 5041
  31. 441
  32. 9
  33. (x + 2)(x + 4)
  34. 784
  35. 3844
  36. 36
  37. (x + 4)(x + 7)
  38. 7056
  39. 101
  40. 6241

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