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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 741 · 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 + 17x + 72? (a) (x + 8)(x + 9) (b) (x − 8)(x − 9) (c) (x + 1)(x + 16) (d) (x + 1)(x + 72)
  2. 2.If a + b = 10 and ab = 9, what is (a − b)²? (a) 100 (b) 64 (c) 82 (d) 8
  3. 3.If a + b = 12 and ab = 27, what is a² + b²? (a) 198 (b) 117 (c) 90 (d) 144
  4. 4.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1089 (b) 909 (c) 729 (d) 1099
  5. 5.What is the product of the roots of x² − 11x + 18 = 0? (a) 17 (b) 19 (c) 18 (d) 11
  6. 6.If a + b = 13 and ab = 30, what is (a − b)²? (a) 49 (b) 169 (c) 7 (d) 109
  7. 7.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 15) (b) (x + 3)(x + 5) (c) (x − 3)(x − 5) (d) (x + 1)(x + 7)
  8. 8.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 5) (b) (x + 2)(x + 4) (c) (x − 2)(x − 4) (d) (x + 1)(x + 8)
  9. 9.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)
  10. 10.What is the sum of the roots of x² − 7x + 10 = 0? (a) 6 (b) 10 (c) 8 (d) 7
  11. 11.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 3146 (b) 3584 (c) 3136 (d) 4096
  12. 12.What are the roots of x² − 9x + 20 = 0? (a) 1 and 8 (b) 1 and 20 (c) 2 and 10 (d) 4 and 5
  13. 13.If a + b = 19 and ab = 84, what is (a − b)²? (a) 25 (b) 193 (c) 361 (d) 5
  14. 14.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x + 1)(x + 56) (c) (x − 7)(x − 8) (d) (x + 1)(x + 14)
  15. 15.If a + b = 10 and ab = 21, what is (a − b)²? (a) 4 (b) 100 (c) 58 (d) 16
  16. 16.What are the roots of x² − 16x + 63 = 0? (a) 1 and 15 (b) 7 and 9 (c) 1 and 63 (d) 3 and 21
  17. 17.If a + b = 13 and ab = 42, what is a² + b²? (a) 253 (b) 127 (c) 85 (d) 169
  18. 18.If a + b = 20 and ab = 99, what is (a − b)²? (a) 2 (b) 202 (c) 400 (d) 4
  19. 19.What are the roots of x² − 8x + 12 = 0? (a) 1 and 7 (b) 3 and 4 (c) 1 and 12 (d) 2 and 6
  20. 20.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 5486 (b) 5476 (c) 4916 (d) 4356
  21. 21.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2116 (b) 2516 (c) 2916 (d) 2926
  22. 22.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 324 (b) 404 (c) 494 (d) 484
  23. 23.If a + b = 14 and ab = 40, what is (a − b)²? (a) 6 (b) 116 (c) 36 (d) 196
  24. 24.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 371 (b) 361 (c) 441 (d) 399
  25. 25.If a + b = 6 and ab = 5, what is a² + b²? (a) 36 (b) 31 (c) 26 (d) 46
  26. 26.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6399 (b) 6251 (c) 6561 (d) 6241
  27. 27.If a + b = 12 and ab = 20, what is a² + b²? (a) 144 (b) 184 (c) 124 (d) 104
  28. 28.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 6) (b) (x − 2)(x − 5) (c) (x + 1)(x + 10) (d) (x + 2)(x + 5)
  29. 29.Which is the correct factorisation of x² + 14x + 48? (a) (x + 6)(x + 8) (b) (x + 1)(x + 48) (c) (x + 1)(x + 13) (d) (x − 6)(x − 8)
  30. 30.Which is the correct factorisation of x² + 12x + 27? (a) (x − 3)(x − 9) (b) (x + 3)(x + 9) (c) (x + 1)(x + 11) (d) (x + 1)(x + 27)
  31. 31.If a + b = 11 and ab = 28, what is (a − b)²? (a) 9 (b) 121 (c) 3 (d) 65
  32. 32.If a + b = 14 and ab = 48, what is (a − b)²? (a) 100 (b) 196 (c) 2 (d) 4
  33. 33.If a + b = 20 and ab = 96, what is (a − b)²? (a) 400 (b) 16 (c) 4 (d) 208
  34. 34.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 7235 (b) 6425 (c) 7225 (d) 5625
  35. 35.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 1024 (b) 784 (c) 794 (d) 896
  36. 36.If a + b = 6 and ab = 8, what is (a − b)²? (a) 4 (b) 2 (c) 20 (d) 36
  37. 37.If a + b = 9 and ab = 20, what is (a − b)²? (a) 1 (b) 41 (c) 61 (d) 81
  38. 38.If a + b = 18 and ab = 80, what is a² + b²? (a) 484 (b) 164 (c) 244 (d) 324
  39. 39.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1681 (b) 1599 (c) 1531 (d) 1521
  40. 40.What is the product of the roots of x² − 13x + 42 = 0? (a) 13 (b) 42 (c) 43 (d) 41

Answer key

  1. (x + 8)(x + 9)
  2. 64
  3. 90
  4. 1089
  5. 18
  6. 49
  7. (x + 3)(x + 5)
  8. (x + 2)(x + 4)
  9. (x + 5)(x + 7)
  10. 7
  11. 3136
  12. 4 and 5
  13. 25
  14. (x + 7)(x + 8)
  15. 16
  16. 7 and 9
  17. 85
  18. 4
  19. 2 and 6
  20. 5476
  21. 2916
  22. 484
  23. 36
  24. 361
  25. 26
  26. 6241
  27. 104
  28. (x + 2)(x + 5)
  29. (x + 6)(x + 8)
  30. (x + 3)(x + 9)
  31. 9
  32. 4
  33. 16
  34. 7225
  35. 784
  36. 4
  37. 1
  38. 164
  39. 1521
  40. 42

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