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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 607 · 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 + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 6)(x + 8) (c) (x + 1)(x + 13) (d) (x + 1)(x + 48)
  2. 2.If a + b = 17 and ab = 72, what is (a − b)²? (a) 289 (b) 145 (c) 1 (d) 217
  3. 3.What is the sum of the roots of x² − 10x + 24 = 0? (a) 24 (b) 11 (c) 10 (d) 9
  4. 4.If a + b = 11 and ab = 24, what is a² + b²? (a) 97 (b) 73 (c) 121 (d) 169
  5. 5.If a + b = 19 and ab = 90, what is a² + b²? (a) 541 (b) 361 (c) 181 (d) 271
  6. 6.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)
  7. 7.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 72 (c) 18 (d) 17
  8. 8.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)
  9. 9.What is the product of the roots of x² − 7x + 12 = 0? (a) 7 (b) 11 (c) 13 (d) 12
  10. 10.If a + b = 7 and ab = 10, what is a² + b²? (a) 49 (b) 69 (c) 39 (d) 29
  11. 11.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 4624 (b) 5184 (c) 4904 (d) 5194
  12. 12.If a + b = 3 and ab = 2, what is (a − b)²? (a) 5 (b) 9 (c) 1 (d) 7
  13. 13.What is the sum of the roots of x² − 12x + 35 = 0? (a) 35 (b) 13 (c) 11 (d) 12
  14. 14.What are the roots of x² − 8x + 12 = 0? (a) 3 and 4 (b) 1 and 12 (c) 2 and 6 (d) 1 and 7
  15. 15.If a + b = 15 and ab = 54, what is a² + b²? (a) 225 (b) 333 (c) 171 (d) 117
  16. 16.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3596 (b) 3844 (c) 3374 (d) 3364
  17. 17.If a + b = 13 and ab = 12, what is a² + b²? (a) 145 (b) 169 (c) 193 (d) 157
  18. 18.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1099 (b) 909 (c) 729 (d) 1089
  19. 19.What are the roots of x² − 7x + 12 = 0? (a) 1 and 6 (b) 2 and 6 (c) 1 and 12 (d) 3 and 4
  20. 20.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 7) (b) (x + 1)(x + 12) (c) (x − 2)(x − 6) (d) (x + 2)(x + 6)
  21. 21.What is the sum of the roots of x² − 15x + 56 = 0? (a) 16 (b) 14 (c) 56 (d) 15
  22. 22.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 1 and 16 (c) 8 and 9 (d) 2 and 36
  23. 23.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 35) (b) (x − 5)(x − 7) (c) (x + 5)(x + 7) (d) (x + 1)(x + 11)
  24. 24.Which is the correct factorisation of x² + 10x + 24? (a) (x − 4)(x − 6) (b) (x + 1)(x + 24) (c) (x + 4)(x + 6) (d) (x + 1)(x + 9)
  25. 25.If a + b = 17 and ab = 60, what is a² + b²? (a) 169 (b) 289 (c) 409 (d) 229
  26. 26.Which is the correct factorisation of x² + 11x + 18? (a) (x − 2)(x − 9) (b) (x + 2)(x + 9) (c) (x + 1)(x + 10) (d) (x + 1)(x + 18)
  27. 27.If a + b = 18 and ab = 80, what is (a − b)²? (a) 324 (b) 2 (c) 164 (d) 4
  28. 28.If a + b = 15 and ab = 56, what is a² + b²? (a) 113 (b) 169 (c) 225 (d) 337
  29. 29.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 6425 (b) 7235 (c) 7225 (d) 5625
  30. 30.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 891 (b) 739 (c) 1089 (d) 729
  31. 31.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 6391 (b) 5929 (c) 6889 (d) 5939
  32. 32.If a + b = 12 and ab = 27, what is a² + b²? (a) 198 (b) 90 (c) 144 (d) 117
  33. 33.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 7569 (b) 8649 (c) 7579 (d) 8091
  34. 34.If a + b = 18 and ab = 72, what is (a − b)²? (a) 6 (b) 36 (c) 180 (d) 324
  35. 35.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9614 (b) 9996 (c) 9604 (d) 10404
  36. 36.What are the roots of x² − 15x + 56 = 0? (a) 1 and 56 (b) 2 and 28 (c) 1 and 14 (d) 7 and 8
  37. 37.If a + b = 14 and ab = 48, what is (a − b)²? (a) 196 (b) 2 (c) 4 (d) 100
  38. 38.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 9984 (b) 9216 (c) 10816 (d) 9226
  39. 39.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 925 (b) 625 (c) 1225 (d) 1235
  40. 40.What is the sum of the roots of x² − 16x + 63 = 0? (a) 16 (b) 63 (c) 17 (d) 15

Answer key

  1. (x + 6)(x + 8)
  2. 1
  3. 10
  4. 73
  5. 181
  6. (x + 6)(x + 9)
  7. 17
  8. (x + 7)(x + 8)
  9. 12
  10. 29
  11. 5184
  12. 1
  13. 12
  14. 2 and 6
  15. 117
  16. 3364
  17. 145
  18. 1089
  19. 3 and 4
  20. (x + 2)(x + 6)
  21. 15
  22. 8 and 9
  23. (x + 5)(x + 7)
  24. (x + 4)(x + 6)
  25. 169
  26. (x + 2)(x + 9)
  27. 4
  28. 113
  29. 7225
  30. 729
  31. 5929
  32. 90
  33. 7569
  34. 36
  35. 9604
  36. 7 and 8
  37. 4
  38. 9216
  39. 1225
  40. 16

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