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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 426 · 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.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)
  2. 2.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 875 (b) 1225 (c) 625 (d) 635
  3. 3.What is the sum of the roots of x² − 8x + 15 = 0? (a) 15 (b) 7 (c) 8 (d) 9
  4. 4.Which is the correct factorisation of x² + 13x + 40? (a) (x − 5)(x − 8) (b) (x + 1)(x + 12) (c) (x + 5)(x + 8) (d) (x + 1)(x + 40)
  5. 5.If a + b = 13 and ab = 30, what is a² + b²? (a) 169 (b) 229 (c) 139 (d) 109
  6. 6.What is the product of the roots of x² − 11x + 28 = 0? (a) 29 (b) 11 (c) 28 (d) 27
  7. 7.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 10) (b) (x − 2)(x − 9) (c) (x + 1)(x + 18) (d) (x + 2)(x + 9)
  8. 8.What are the roots of x² − 16x + 63 = 0? (a) 1 and 15 (b) 3 and 21 (c) 7 and 9 (d) 1 and 63
  9. 9.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 24) (b) (x + 1)(x + 10) (c) (x − 3)(x − 8) (d) (x + 3)(x + 8)
  10. 10.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6561 (b) 6241 (c) 6251 (d) 6399
  11. 11.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 784 (b) 1024 (c) 1034 (d) 904
  12. 12.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 3584 (b) 3136 (c) 3146 (d) 4096
  13. 13.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 916 (b) 676 (c) 1166 (d) 1156
  14. 14.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 4875 (b) 4235 (c) 5625 (d) 4225
  15. 15.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 18 (c) 16 (d) 17
  16. 16.Which is the correct factorisation of x² + 6x + 8? (a) (x − 2)(x − 4) (b) (x + 2)(x + 4) (c) (x + 1)(x + 5) (d) (x + 1)(x + 8)
  17. 17.If a + b = 20 and ab = 99, what is (a − b)²? (a) 2 (b) 202 (c) 400 (d) 4
  18. 18.If a + b = 13 and ab = 40, what is (a − b)²? (a) 9 (b) 89 (c) 169 (d) 3
  19. 19.If a + b = 13 and ab = 12, what is a² + b²? (a) 145 (b) 193 (c) 157 (d) 169
  20. 20.What are the roots of x² − 9x + 18 = 0? (a) 2 and 9 (b) 3 and 6 (c) 1 and 18 (d) 1 and 8
  21. 21.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 6) (b) (x + 3)(x + 4) (c) (x − 3)(x − 4) (d) (x + 1)(x + 12)
  22. 22.If a + b = 10 and ab = 24, what is a² + b²? (a) 148 (b) 76 (c) 100 (d) 52
  23. 23.If a + b = 20 and ab = 99, what is a² + b²? (a) 202 (b) 400 (c) 301 (d) 598
  24. 24.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x + 6)(x + 7) (c) (x + 1)(x + 42) (d) (x − 6)(x − 7)
  25. 25.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2714 (b) 2504 (c) 2704 (d) 2304
  26. 26.If a + b = 14 and ab = 33, what is a² + b²? (a) 262 (b) 196 (c) 163 (d) 130
  27. 27.What is the sum of the roots of x² − 14x + 48 = 0? (a) 14 (b) 48 (c) 13 (d) 15
  28. 28.What is the sum of the roots of x² − 12x + 32 = 0? (a) 11 (b) 12 (c) 32 (d) 13
  29. 29.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 2025 (b) 2525 (c) 3025 (d) 3035
  30. 30.What is the sum of the roots of x² − 8x + 12 = 0? (a) 7 (b) 8 (c) 12 (d) 9
  31. 31.If a + b = 13 and ab = 42, what is (a − b)²? (a) 169 (b) 85 (c) 127 (d) 1
  32. 32.What are the roots of x² − 15x + 56 = 0? (a) 1 and 56 (b) 1 and 14 (c) 7 and 8 (d) 2 and 28
  33. 33.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 891 (b) 1089 (c) 729 (d) 739
  34. 34.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8474 (b) 8464 (c) 7744 (d) 8104
  35. 35.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 5786 (b) 5776 (c) 7056 (d) 6384
  36. 36.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x − 7)(x − 8) (c) (x + 7)(x + 8) (d) (x + 1)(x + 56)
  37. 37.If a + b = 14 and ab = 48, what is (a − b)²? (a) 100 (b) 2 (c) 4 (d) 196
  38. 38.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1296 (b) 1946 (c) 1616 (d) 1936
  39. 39.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 441 (b) 399 (c) 371 (d) 361
  40. 40.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2209 (b) 2809 (c) 2491 (d) 2219

Answer key

  1. (x + 8)(x + 9)
  2. 625
  3. 8
  4. (x + 5)(x + 8)
  5. 109
  6. 28
  7. (x + 2)(x + 9)
  8. 7 and 9
  9. (x + 3)(x + 8)
  10. 6241
  11. 1024
  12. 3136
  13. 1156
  14. 4225
  15. 17
  16. (x + 2)(x + 4)
  17. 4
  18. 9
  19. 145
  20. 3 and 6
  21. (x + 3)(x + 4)
  22. 52
  23. 202
  24. (x + 6)(x + 7)
  25. 2704
  26. 130
  27. 14
  28. 12
  29. 3025
  30. 8
  31. 1
  32. 7 and 8
  33. 729
  34. 8464
  35. 5776
  36. (x + 7)(x + 8)
  37. 4
  38. 1936
  39. 361
  40. 2209

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