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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 849 · 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 = 12 and ab = 27, what is (a − b)²? (a) 144 (b) 90 (c) 6 (d) 36
  2. 2.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4366 (b) 4356 (c) 5476 (d) 4884
  3. 3.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6094 (b) 6724 (c) 6084 (d) 6396
  4. 4.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 7)(x + 9) (c) (x − 7)(x − 9) (d) (x + 1)(x + 63)
  5. 5.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x − 6)(x − 7) (c) (x + 1)(x + 12) (d) (x + 6)(x + 7)
  6. 6.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8464 (b) 7744 (c) 8474 (d) 8104
  7. 7.If a + b = 16 and ab = 60, what is (a − b)²? (a) 136 (b) 256 (c) 16 (d) 4
  8. 8.If a + b = 20 and ab = 96, what is (a − b)²? (a) 4 (b) 400 (c) 208 (d) 16
  9. 9.If a + b = 14 and ab = 33, what is (a − b)²? (a) 196 (b) 8 (c) 64 (d) 130
  10. 10.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2926 (b) 2916 (c) 2516 (d) 2116
  11. 11.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x + 7)(x + 8) (c) (x + 1)(x + 56) (d) (x − 7)(x − 8)
  12. 12.If a + b = 19 and ab = 88, what is a² + b²? (a) 361 (b) 537 (c) 185 (d) 273
  13. 13.If a + b = 21 and ab = 108, what is a² + b²? (a) 225 (b) 333 (c) 441 (d) 657
  14. 14.If a + b = 17 and ab = 72, what is a² + b²? (a) 433 (b) 217 (c) 289 (d) 145
  15. 15.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 17 (c) 73 (d) 72
  16. 16.If a + b = 10 and ab = 16, what is (a − b)²? (a) 100 (b) 6 (c) 36 (d) 68
  17. 17.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 909 (b) 729 (c) 1099 (d) 1089
  18. 18.If a + b = 15 and ab = 36, what is (a − b)²? (a) 9 (b) 81 (c) 225 (d) 153
  19. 19.If a + b = 6 and ab = 8, what is (a − b)²? (a) 2 (b) 36 (c) 20 (d) 4
  20. 20.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1849 (b) 1369 (c) 1379 (d) 1591
  21. 21.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 256 (b) 586 (c) 416 (d) 576
  22. 22.What are the roots of x² − 12x + 35 = 0? (a) 1 and 35 (b) 2 and 10 (c) 5 and 7 (d) 1 and 11
  23. 23.What are the roots of x² − 6x + 8 = 0? (a) 1 and 8 (b) 1 and 5 (c) 2 and 4 (d) 2 and 5
  24. 24.What are the roots of x² − 12x + 27 = 0? (a) 3 and 9 (b) 1 and 27 (c) 1 and 11 (d) 2 and 10
  25. 25.Which is the correct factorisation of x² + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 1)(x + 10) (c) (x + 1)(x + 30) (d) (x + 5)(x + 6)
  26. 26.Which is the correct factorisation of x² + 9x + 14? (a) (x − 2)(x − 7) (b) (x + 1)(x + 14) (c) (x + 2)(x + 7) (d) (x + 1)(x + 8)
  27. 27.If a + b = 9 and ab = 20, what is (a − b)²? (a) 81 (b) 61 (c) 41 (d) 1
  28. 28.What are the roots of x² − 11x + 18 = 0? (a) 3 and 6 (b) 1 and 10 (c) 1 and 18 (d) 2 and 9
  29. 29.What is the sum of the roots of x² − 15x + 56 = 0? (a) 14 (b) 15 (c) 56 (d) 16
  30. 30.If a + b = 11 and ab = 24, what is a² + b²? (a) 121 (b) 97 (c) 73 (d) 169
  31. 31.If a + b = 10 and ab = 16, what is a² + b²? (a) 132 (b) 100 (c) 68 (d) 84
  32. 32.What are the roots of x² − 13x + 42 = 0? (a) 1 and 42 (b) 6 and 7 (c) 1 and 12 (d) 2 and 21
  33. 33.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 8)(x + 9) (c) (x − 8)(x − 9) (d) (x + 1)(x + 16)
  34. 34.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x + 1)(x + 48) (c) (x − 6)(x − 8) (d) (x + 6)(x + 8)
  35. 35.If a + b = 17 and ab = 70, what is (a − b)²? (a) 149 (b) 9 (c) 3 (d) 289
  36. 36.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2314 (b) 2704 (c) 2304 (d) 2496
  37. 37.What is the sum of the roots of x² − 12x + 35 = 0? (a) 11 (b) 35 (c) 13 (d) 12
  38. 38.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 5625 (b) 5635 (c) 4225 (d) 4925
  39. 39.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1599 (b) 1531 (c) 1521 (d) 1681
  40. 40.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 18) (b) (x − 3)(x − 6) (c) (x + 1)(x + 8) (d) (x + 3)(x + 6)

Answer key

  1. 36
  2. 4356
  3. 6084
  4. (x + 7)(x + 9)
  5. (x + 6)(x + 7)
  6. 8464
  7. 16
  8. 16
  9. 64
  10. 2916
  11. (x + 7)(x + 8)
  12. 185
  13. 225
  14. 145
  15. 72
  16. 36
  17. 1089
  18. 81
  19. 4
  20. 1369
  21. 576
  22. 5 and 7
  23. 2 and 4
  24. 3 and 9
  25. (x + 5)(x + 6)
  26. (x + 2)(x + 7)
  27. 1
  28. 2 and 9
  29. 15
  30. 73
  31. 68
  32. 6 and 7
  33. (x + 8)(x + 9)
  34. (x + 6)(x + 8)
  35. 9
  36. 2304
  37. 12
  38. 5625
  39. 1521
  40. (x + 3)(x + 6)

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