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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 714 · 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 + 8x + 12? (a) (x + 1)(x + 7) (b) (x − 2)(x − 6) (c) (x + 1)(x + 12) (d) (x + 2)(x + 6)
  2. 2.If a + b = 21 and ab = 108, what is a² + b²? (a) 657 (b) 441 (c) 225 (d) 333
  3. 3.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 9801 (b) 10001 (c) 10211 (d) 10201
  4. 4.What is the sum of the roots of x² − 12x + 32 = 0? (a) 12 (b) 13 (c) 11 (d) 32
  5. 5.If a + b = 13 and ab = 12, what is a² + b²? (a) 193 (b) 157 (c) 145 (d) 169
  6. 6.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 17 (c) 18 (d) 16
  7. 7.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 4235 (b) 4225 (c) 5625 (d) 4875
  8. 8.If a + b = 15 and ab = 36, what is (a − b)²? (a) 225 (b) 9 (c) 153 (d) 81
  9. 9.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 494 (b) 404 (c) 324 (d) 484
  10. 10.If a + b = 13 and ab = 36, what is (a − b)²? (a) 25 (b) 97 (c) 169 (d) 5
  11. 11.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6396 (b) 6724 (c) 6084 (d) 6094
  12. 12.If a + b = 10 and ab = 21, what is a² + b²? (a) 58 (b) 79 (c) 142 (d) 100
  13. 13.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 4) (b) (x + 1)(x + 6) (c) (x − 2)(x − 3) (d) (x + 2)(x + 3)
  14. 14.If a + b = 11 and ab = 28, what is (a − b)²? (a) 65 (b) 9 (c) 121 (d) 3
  15. 15.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 72 (c) 73 (d) 71
  16. 16.What is the sum of the roots of x² − 15x + 56 = 0? (a) 14 (b) 56 (c) 16 (d) 15
  17. 17.What are the roots of x² − 13x + 40 = 0? (a) 1 and 12 (b) 1 and 40 (c) 5 and 8 (d) 2 and 20
  18. 18.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x + 1)(x + 63) (c) (x + 1)(x + 15) (d) (x − 7)(x − 9)
  19. 19.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 8)(x + 9) (c) (x + 1)(x + 16) (d) (x − 8)(x − 9)
  20. 20.What is the sum of the roots of x² − 7x + 12 = 0? (a) 8 (b) 12 (c) 6 (d) 7
  21. 21.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 676 (b) 916 (c) 1156 (d) 1166
  22. 22.If a + b = 17 and ab = 70, what is a² + b²? (a) 429 (b) 219 (c) 289 (d) 149
  23. 23.What are the roots of x² − 15x + 54 = 0? (a) 2 and 27 (b) 1 and 14 (c) 1 and 54 (d) 6 and 9
  24. 24.What is the sum of the roots of x² − 12x + 35 = 0? (a) 11 (b) 35 (c) 13 (d) 12
  25. 25.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 45) (b) (x + 5)(x + 9) (c) (x − 5)(x − 9) (d) (x + 1)(x + 13)
  26. 26.If a + b = 7 and ab = 6, what is a² + b²? (a) 37 (b) 61 (c) 49 (d) 43
  27. 27.If a + b = 18 and ab = 72, what is a² + b²? (a) 468 (b) 252 (c) 180 (d) 324
  28. 28.What is the product of the roots of x² − 11x + 30 = 0? (a) 29 (b) 30 (c) 11 (d) 31
  29. 29.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)
  30. 30.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 1)(x + 14) (c) (x − 7)(x − 8) (d) (x + 7)(x + 8)
  31. 31.If a + b = 16 and ab = 55, what is a² + b²? (a) 201 (b) 146 (c) 256 (d) 366
  32. 32.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 7066 (b) 6416 (c) 7056 (d) 5776
  33. 33.Which is the correct factorisation of x² + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 5)(x + 6) (c) (x + 1)(x + 10) (d) (x + 1)(x + 30)
  34. 34.What is the product of the roots of x² − 14x + 48 = 0? (a) 47 (b) 14 (c) 49 (d) 48
  35. 35.What is the sum of the roots of x² − 9x + 14 = 0? (a) 10 (b) 8 (c) 9 (d) 14
  36. 36.What are the roots of x² − 10x + 16 = 0? (a) 1 and 9 (b) 3 and 7 (c) 1 and 16 (d) 2 and 8
  37. 37.Which is the correct factorisation of x² + 10x + 16? (a) (x + 2)(x + 8) (b) (x + 1)(x + 16) (c) (x − 2)(x − 8) (d) (x + 1)(x + 9)
  38. 38.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 784 (b) 794 (c) 1024 (d) 896
  39. 39.If a + b = 13 and ab = 22, what is (a − b)²? (a) 81 (b) 169 (c) 9 (d) 125
  40. 40.What are the roots of x² − 16x + 63 = 0? (a) 7 and 9 (b) 1 and 63 (c) 3 and 21 (d) 1 and 15

Answer key

  1. (x + 2)(x + 6)
  2. 225
  3. 10201
  4. 12
  5. 145
  6. 17
  7. 4225
  8. 81
  9. 484
  10. 25
  11. 6084
  12. 58
  13. (x + 2)(x + 3)
  14. 9
  15. 72
  16. 15
  17. 5 and 8
  18. (x + 7)(x + 9)
  19. (x + 8)(x + 9)
  20. 7
  21. 1156
  22. 149
  23. 6 and 9
  24. 12
  25. (x + 5)(x + 9)
  26. 37
  27. 180
  28. 30
  29. (x + 5)(x + 7)
  30. (x + 7)(x + 8)
  31. 146
  32. 7056
  33. (x + 5)(x + 6)
  34. 48
  35. 9
  36. 2 and 8
  37. (x + 2)(x + 8)
  38. 784
  39. 81
  40. 7 and 9

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