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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 82 · 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 = 13 and ab = 42, what is (a − b)²? (a) 85 (b) 1 (c) 127 (d) 169
  2. 2.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 7) (b) (x + 2)(x + 6) (c) (x − 2)(x − 6) (d) (x + 1)(x + 12)
  3. 3.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)
  4. 4.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 5329 (b) 4489 (c) 4909 (d) 5339
  5. 5.What is the product of the roots of x² − 13x + 40 = 0? (a) 40 (b) 13 (c) 41 (d) 39
  6. 6.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 9) (b) (x + 1)(x + 24) (c) (x + 4)(x + 6) (d) (x − 4)(x − 6)
  7. 7.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 404 (b) 324 (c) 484 (d) 494
  8. 8.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 32) (b) (x + 4)(x + 8) (c) (x − 4)(x − 8) (d) (x + 1)(x + 11)
  9. 9.What is the product of the roots of x² − 15x + 56 = 0? (a) 57 (b) 15 (c) 55 (d) 56
  10. 10.What is the sum of the roots of x² − 16x + 63 = 0? (a) 16 (b) 63 (c) 15 (d) 17
  11. 11.If a + b = 19 and ab = 90, what is (a − b)²? (a) 271 (b) 181 (c) 1 (d) 361
  12. 12.What is the sum of the roots of x² − 8x + 12 = 0? (a) 12 (b) 8 (c) 9 (d) 7
  13. 13.If a + b = 21 and ab = 108, what is a² + b²? (a) 657 (b) 441 (c) 333 (d) 225
  14. 14.If a + b = 9 and ab = 14, what is a² + b²? (a) 109 (b) 67 (c) 53 (d) 81
  15. 15.If a + b = 11 and ab = 28, what is (a − b)²? (a) 65 (b) 9 (c) 121 (d) 3
  16. 16.If a + b = 15 and ab = 36, what is a² + b²? (a) 225 (b) 189 (c) 297 (d) 153
  17. 17.Which is the correct factorisation of x² + 5x + 6? (a) (x − 2)(x − 3) (b) (x + 1)(x + 6) (c) (x + 1)(x + 4) (d) (x + 2)(x + 3)
  18. 18.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 40) (b) (x − 5)(x − 8) (c) (x + 5)(x + 8) (d) (x + 1)(x + 12)
  19. 19.What is the product of the roots of x² − 12x + 35 = 0? (a) 12 (b) 35 (c) 36 (d) 34
  20. 20.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9996 (b) 9614 (c) 9604 (d) 10404
  21. 21.If a + b = 14 and ab = 33, what is a² + b²? (a) 262 (b) 163 (c) 196 (d) 130
  22. 22.If a + b = 19 and ab = 90, what is a² + b²? (a) 361 (b) 181 (c) 541 (d) 271
  23. 23.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 401 (b) 361 (c) 451 (d) 441
  24. 24.If a + b = 16 and ab = 60, what is (a − b)²? (a) 136 (b) 4 (c) 16 (d) 256
  25. 25.If a + b = 18 and ab = 80, what is a² + b²? (a) 244 (b) 484 (c) 324 (d) 164
  26. 26.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 71 (c) 72 (d) 17
  27. 27.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3364 (b) 3854 (c) 3844 (d) 3604
  28. 28.If a + b = 12 and ab = 20, what is a² + b²? (a) 184 (b) 144 (c) 124 (d) 104
  29. 29.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1599 (b) 1521 (c) 1681 (d) 1531
  30. 30.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 14) (c) (x + 1)(x + 56) (d) (x + 7)(x + 8)
  31. 31.If a + b = 17 and ab = 72, what is (a − b)²? (a) 145 (b) 289 (c) 1 (d) 217
  32. 32.If a + b = 13 and ab = 12, what is a² + b²? (a) 145 (b) 169 (c) 157 (d) 193
  33. 33.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 16 (c) 17 (d) 18
  34. 34.If a + b = 14 and ab = 45, what is (a − b)²? (a) 4 (b) 196 (c) 106 (d) 16
  35. 35.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10004 (b) 10414 (c) 9604 (d) 10404
  36. 36.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x + 1)(x + 10) (c) (x + 5)(x + 6) (d) (x − 5)(x − 6)
  37. 37.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 8096 (b) 7744 (c) 8464 (d) 7754
  38. 38.What are the roots of x² − 8x + 15 = 0? (a) 1 and 7 (b) 3 and 5 (c) 2 and 6 (d) 1 and 15
  39. 39.If a + b = 15 and ab = 36, what is (a − b)²? (a) 153 (b) 9 (c) 225 (d) 81
  40. 40.If a + b = 14 and ab = 24, what is (a − b)²? (a) 196 (b) 148 (c) 100 (d) 10

Answer key

  1. 1
  2. (x + 2)(x + 6)
  3. (x + 8)(x + 9)
  4. 5329
  5. 40
  6. (x + 4)(x + 6)
  7. 484
  8. (x + 4)(x + 8)
  9. 56
  10. 16
  11. 1
  12. 8
  13. 225
  14. 53
  15. 9
  16. 153
  17. (x + 2)(x + 3)
  18. (x + 5)(x + 8)
  19. 35
  20. 9604
  21. 130
  22. 181
  23. 441
  24. 16
  25. 164
  26. 72
  27. 3844
  28. 104
  29. 1521
  30. (x + 7)(x + 8)
  31. 1
  32. 145
  33. 17
  34. 16
  35. 10404
  36. (x + 5)(x + 6)
  37. 7744
  38. 3 and 5
  39. 81
  40. 100

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