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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 375 · 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.If a + b = 9 and ab = 20, what is a² + b²? (a) 41 (b) 81 (c) 121 (d) 61
  2. 2.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)
  3. 3.Which is the correct factorisation of x² + 15x + 54? (a) (x − 6)(x − 9) (b) (x + 1)(x + 54) (c) (x + 6)(x + 9) (d) (x + 1)(x + 14)
  4. 4.Which is the correct factorisation of x² + 11x + 24? (a) (x + 3)(x + 8) (b) (x + 1)(x + 10) (c) (x − 3)(x − 8) (d) (x + 1)(x + 24)
  5. 5.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 17 (c) 18 (d) 72
  6. 6.If a + b = 13 and ab = 12, what is a² + b²? (a) 145 (b) 193 (c) 157 (d) 169
  7. 7.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x − 5)(x − 7) (c) (x + 1)(x + 35) (d) (x + 5)(x + 7)
  8. 8.If a + b = 16 and ab = 60, what is a² + b²? (a) 256 (b) 376 (c) 196 (d) 136
  9. 9.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 9) (b) (x + 1)(x + 21) (c) (x − 3)(x − 7) (d) (x + 3)(x + 7)
  10. 10.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 5625 (b) 4235 (c) 4225 (d) 4875
  11. 11.Which is the correct factorisation of x² + 12x + 32? (a) (x − 4)(x − 8) (b) (x + 1)(x + 11) (c) (x + 1)(x + 32) (d) (x + 4)(x + 8)
  12. 12.What is the product of the roots of x² − 12x + 32 = 0? (a) 33 (b) 32 (c) 12 (d) 31
  13. 13.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 7921 (b) 8281 (c) 8101 (d) 8291
  14. 14.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 9025 (b) 7225 (c) 8075 (d) 7235
  15. 15.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 9801 (b) 9999 (c) 10201 (d) 9811
  16. 16.Which is the correct factorisation of x² + 9x + 20? (a) (x − 4)(x − 5) (b) (x + 1)(x + 8) (c) (x + 4)(x + 5) (d) (x + 1)(x + 20)
  17. 17.What is the sum of the roots of x² − 16x + 63 = 0? (a) 63 (b) 17 (c) 16 (d) 15
  18. 18.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)
  19. 19.If a + b = 14 and ab = 24, what is a² + b²? (a) 244 (b) 148 (c) 196 (d) 172
  20. 20.If a + b = 20 and ab = 96, what is (a − b)²? (a) 4 (b) 16 (c) 208 (d) 400
  21. 21.What are the roots of x² − 14x + 48 = 0? (a) 1 and 48 (b) 2 and 24 (c) 6 and 8 (d) 1 and 13
  22. 22.If a + b = 4 and ab = 3, what is (a − b)²? (a) 16 (b) 4 (c) 2 (d) 10
  23. 23.If a + b = 17 and ab = 70, what is (a − b)²? (a) 289 (b) 149 (c) 9 (d) 3
  24. 24.What is the product of the roots of x² − 11x + 24 = 0? (a) 11 (b) 23 (c) 25 (d) 24
  25. 25.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 5625 (b) 6375 (c) 5635 (d) 7225
  26. 26.If a + b = 3 and ab = 2, what is (a − b)²? (a) 7 (b) 9 (c) 1 (d) 5
  27. 27.If a + b = 17 and ab = 70, what is a² + b²? (a) 289 (b) 149 (c) 429 (d) 219
  28. 28.Which is the correct factorisation of x² + 8x + 15? (a) (x + 3)(x + 5) (b) (x + 1)(x + 7) (c) (x + 1)(x + 15) (d) (x − 3)(x − 5)
  29. 29.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 5) (b) (x + 1)(x + 8) (c) (x + 2)(x + 4) (d) (x − 2)(x − 4)
  30. 30.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x + 7)(x + 9) (c) (x − 7)(x − 9) (d) (x + 1)(x + 15)
  31. 31.If a + b = 13 and ab = 36, what is (a − b)²? (a) 97 (b) 25 (c) 169 (d) 5
  32. 32.If a + b = 13 and ab = 40, what is a² + b²? (a) 249 (b) 129 (c) 89 (d) 169
  33. 33.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 5776 (b) 7056 (c) 6384 (d) 5786
  34. 34.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8109 (b) 7569 (c) 8659 (d) 8649
  35. 35.If a + b = 9 and ab = 14, what is (a − b)²? (a) 53 (b) 25 (c) 5 (d) 81
  36. 36.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 6)(x + 8) (c) (x + 1)(x + 13) (d) (x + 1)(x + 48)
  37. 37.If a + b = 8 and ab = 15, what is a² + b²? (a) 64 (b) 49 (c) 94 (d) 34
  38. 38.If a + b = 8 and ab = 7, what is (a − b)²? (a) 50 (b) 6 (c) 36 (d) 64
  39. 39.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3979 (b) 3609 (c) 3969 (d) 3249
  40. 40.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3604 (b) 3854 (c) 3844 (d) 3364

Answer key

  1. 41
  2. (x + 2)(x + 3)
  3. (x + 6)(x + 9)
  4. (x + 3)(x + 8)
  5. 17
  6. 145
  7. (x + 5)(x + 7)
  8. 136
  9. (x + 3)(x + 7)
  10. 4225
  11. (x + 4)(x + 8)
  12. 32
  13. 8281
  14. 7225
  15. 9801
  16. (x + 4)(x + 5)
  17. 16
  18. (x + 8)(x + 9)
  19. 148
  20. 16
  21. 6 and 8
  22. 4
  23. 9
  24. 24
  25. 5625
  26. 1
  27. 149
  28. (x + 3)(x + 5)
  29. (x + 2)(x + 4)
  30. (x + 7)(x + 9)
  31. 25
  32. 89
  33. 5776
  34. 8649
  35. 25
  36. (x + 6)(x + 8)
  37. 34
  38. 36
  39. 3969
  40. 3844

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