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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 514 · 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 + 1)(x + 72) (b) (x + 1)(x + 16) (c) (x − 8)(x − 9) (d) (x + 8)(x + 9)
  2. 2.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 7921 (b) 8101 (c) 8281 (d) 8291
  3. 3.If a + b = 13 and ab = 30, what is a² + b²? (a) 139 (b) 229 (c) 109 (d) 169
  4. 4.What is the product of the roots of x² − 7x + 12 = 0? (a) 13 (b) 12 (c) 7 (d) 11
  5. 5.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6399 (b) 6251 (c) 6241 (d) 6561
  6. 6.If a + b = 17 and ab = 60, what is (a − b)²? (a) 169 (b) 7 (c) 289 (d) 49
  7. 7.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 784 (b) 1024 (c) 1034 (d) 904
  8. 8.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 916 (b) 676 (c) 1166 (d) 1156
  9. 9.What is the sum of the roots of x² − 6x + 8 = 0? (a) 8 (b) 6 (c) 5 (d) 7
  10. 10.If a + b = 21 and ab = 108, what is a² + b²? (a) 657 (b) 441 (c) 333 (d) 225
  11. 11.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 9) (b) (x + 1)(x + 16) (c) (x + 2)(x + 8) (d) (x − 2)(x − 8)
  12. 12.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 17 (c) 73 (d) 72
  13. 13.If a + b = 19 and ab = 90, what is a² + b²? (a) 541 (b) 361 (c) 271 (d) 181
  14. 14.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)
  15. 15.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 371 (b) 361 (c) 441 (d) 399
  16. 16.If a + b = 8 and ab = 7, what is a² + b²? (a) 78 (b) 50 (c) 57 (d) 64
  17. 17.Which is the correct factorisation of x² + 6x + 8? (a) (x + 2)(x + 4) (b) (x + 1)(x + 8) (c) (x − 2)(x − 4) (d) (x + 1)(x + 5)
  18. 18.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 6)(x + 7) (c) (x + 1)(x + 12) (d) (x + 1)(x + 42)
  19. 19.If a + b = 17 and ab = 70, what is (a − b)²? (a) 289 (b) 3 (c) 149 (d) 9
  20. 20.What is the product of the roots of x² − 15x + 54 = 0? (a) 15 (b) 53 (c) 55 (d) 54
  21. 21.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 794 (b) 896 (c) 784 (d) 1024
  22. 22.If a + b = 13 and ab = 40, what is a² + b²? (a) 169 (b) 129 (c) 249 (d) 89
  23. 23.If a + b = 14 and ab = 45, what is a² + b²? (a) 151 (b) 286 (c) 106 (d) 196
  24. 24.What are the roots of x² − 10x + 16 = 0? (a) 1 and 16 (b) 1 and 9 (c) 3 and 7 (d) 2 and 8
  25. 25.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 5339 (b) 4489 (c) 4909 (d) 5329
  26. 26.What are the roots of x² − 11x + 30 = 0? (a) 1 and 30 (b) 1 and 10 (c) 5 and 6 (d) 2 and 15
  27. 27.What is the sum of the roots of x² − 12x + 32 = 0? (a) 32 (b) 13 (c) 11 (d) 12
  28. 28.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 576 (b) 416 (c) 586 (d) 256
  29. 29.If a + b = 15 and ab = 44, what is (a − b)²? (a) 7 (b) 225 (c) 137 (d) 49
  30. 30.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)
  31. 31.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 32) (b) (x + 4)(x + 8) (c) (x + 1)(x + 11) (d) (x − 4)(x − 8)
  32. 32.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2704 (b) 2714 (c) 2304 (d) 2504
  33. 33.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 6) (b) (x − 2)(x − 5) (c) (x + 1)(x + 10) (d) (x + 2)(x + 5)
  34. 34.What is the product of the roots of x² − 13x + 42 = 0? (a) 41 (b) 43 (c) 13 (d) 42
  35. 35.Which is the correct factorisation of x² + 14x + 45? (a) (x − 5)(x − 9) (b) (x + 5)(x + 9) (c) (x + 1)(x + 45) (d) (x + 1)(x + 13)
  36. 36.If a + b = 13 and ab = 22, what is a² + b²? (a) 147 (b) 169 (c) 213 (d) 125
  37. 37.If a + b = 6 and ab = 5, what is (a − b)²? (a) 4 (b) 16 (c) 26 (d) 36
  38. 38.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 729 (b) 891 (c) 1089 (d) 739
  39. 39.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 2035 (b) 1225 (c) 2025 (d) 1625
  40. 40.What is the sum of the roots of x² − 16x + 63 = 0? (a) 63 (b) 15 (c) 17 (d) 16

Answer key

  1. (x + 8)(x + 9)
  2. 8281
  3. 109
  4. 12
  5. 6241
  6. 49
  7. 1024
  8. 1156
  9. 6
  10. 225
  11. (x + 2)(x + 8)
  12. 72
  13. 181
  14. (x + 2)(x + 6)
  15. 361
  16. 50
  17. (x + 2)(x + 4)
  18. (x + 6)(x + 7)
  19. 9
  20. 54
  21. 784
  22. 89
  23. 106
  24. 2 and 8
  25. 5329
  26. 5 and 6
  27. 12
  28. 576
  29. 49
  30. (x + 7)(x + 9)
  31. (x + 4)(x + 8)
  32. 2704
  33. (x + 2)(x + 5)
  34. 42
  35. (x + 5)(x + 9)
  36. 125
  37. 16
  38. 729
  39. 2025
  40. 16

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