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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 644 · 40 questions
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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.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 18 (c) 17 (d) 16
  2. 2.Which is the correct factorisation of x² + 8x + 12? (a) (x − 2)(x − 6) (b) (x + 1)(x + 12) (c) (x + 1)(x + 7) (d) (x + 2)(x + 6)
  3. 3.What are the roots of x² − 11x + 24 = 0? (a) 1 and 10 (b) 3 and 8 (c) 2 and 12 (d) 1 and 24
  4. 4.Which is the correct factorisation of x² + 7x + 10? (a) (x − 2)(x − 5) (b) (x + 1)(x + 10) (c) (x + 1)(x + 6) (d) (x + 2)(x + 5)
  5. 5.If a + b = 7 and ab = 6, what is a² + b²? (a) 49 (b) 61 (c) 37 (d) 43
  6. 6.What is the product of the roots of x² − 12x + 32 = 0? (a) 32 (b) 31 (c) 12 (d) 33
  7. 7.Which is the correct factorisation of x² + 14x + 48? (a) (x + 6)(x + 8) (b) (x + 1)(x + 13) (c) (x + 1)(x + 48) (d) (x − 6)(x − 8)
  8. 8.What is the product of the roots of x² − 8x + 12 = 0? (a) 12 (b) 13 (c) 11 (d) 8
  9. 9.Which is the correct factorisation of x² + 13x + 40? (a) (x − 5)(x − 8) (b) (x + 1)(x + 12) (c) (x + 5)(x + 8) (d) (x + 1)(x + 40)
  10. 10.What are the roots of x² − 13x + 42 = 0? (a) 2 and 21 (b) 1 and 12 (c) 6 and 7 (d) 1 and 42
  11. 11.What are the roots of x² − 11x + 30 = 0? (a) 1 and 30 (b) 2 and 15 (c) 1 and 10 (d) 5 and 6
  12. 12.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 576 (b) 416 (c) 586 (d) 256
  13. 13.If a + b = 8 and ab = 7, what is a² + b²? (a) 64 (b) 78 (c) 57 (d) 50
  14. 14.What is the product of the roots of x² − 5x + 6 = 0? (a) 8 (b) 5 (c) 6 (d) 7
  15. 15.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1454 (b) 1596 (c) 1764 (d) 1444
  16. 16.If a + b = 14 and ab = 33, what is (a − b)²? (a) 64 (b) 8 (c) 196 (d) 130
  17. 17.If a + b = 18 and ab = 80, what is (a − b)²? (a) 2 (b) 324 (c) 164 (d) 4
  18. 18.What is the sum of the roots of x² − 14x + 48 = 0? (a) 14 (b) 48 (c) 13 (d) 15
  19. 19.What is the product of the roots of x² − 15x + 56 = 0? (a) 57 (b) 55 (c) 15 (d) 56
  20. 20.What is the sum of the roots of x² − 13x + 40 = 0? (a) 14 (b) 13 (c) 12 (d) 40
  21. 21.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1531 (b) 1681 (c) 1521 (d) 1599
  22. 22.What are the roots of x² − 7x + 10 = 0? (a) 3 and 4 (b) 1 and 10 (c) 2 and 5 (d) 1 and 6
  23. 23.If a + b = 9 and ab = 18, what is (a − b)²? (a) 45 (b) 81 (c) 9 (d) 3
  24. 24.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10404 (b) 9604 (c) 10004 (d) 10414
  25. 25.What is the product of the roots of x² − 11x + 18 = 0? (a) 11 (b) 19 (c) 18 (d) 17
  26. 26.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8474 (b) 8104 (c) 7744 (d) 8464
  27. 27.If a + b = 18 and ab = 72, what is (a − b)²? (a) 180 (b) 6 (c) 324 (d) 36
  28. 28.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 404 (b) 484 (c) 494 (d) 324
  29. 29.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 539 (b) 289 (c) 409 (d) 529
  30. 30.If a + b = 13 and ab = 36, what is a² + b²? (a) 97 (b) 169 (c) 133 (d) 241
  31. 31.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 8125 (b) 9025 (c) 7225 (d) 9035
  32. 32.What are the roots of x² − 10x + 24 = 0? (a) 1 and 24 (b) 2 and 12 (c) 1 and 9 (d) 4 and 6
  33. 33.If a + b = 15 and ab = 36, what is (a − b)²? (a) 9 (b) 153 (c) 81 (d) 225
  34. 34.Which is the correct factorisation of x² + 10x + 21? (a) (x + 3)(x + 7) (b) (x + 1)(x + 9) (c) (x + 1)(x + 21) (d) (x − 3)(x − 7)
  35. 35.If a + b = 19 and ab = 90, what is a² + b²? (a) 361 (b) 541 (c) 181 (d) 271
  36. 36.If a + b = 19 and ab = 88, what is a² + b²? (a) 273 (b) 537 (c) 361 (d) 185
  37. 37.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x − 5)(x − 7) (c) (x + 5)(x + 7) (d) (x + 1)(x + 35)
  38. 38.If a + b = 13 and ab = 42, what is a² + b²? (a) 253 (b) 169 (c) 127 (d) 85
  39. 39.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1936 (b) 1306 (c) 1584 (d) 1296
  40. 40.If a + b = 13 and ab = 36, what is (a − b)²? (a) 169 (b) 5 (c) 25 (d) 97

Answer key

  1. 17
  2. (x + 2)(x + 6)
  3. 3 and 8
  4. (x + 2)(x + 5)
  5. 37
  6. 32
  7. (x + 6)(x + 8)
  8. 12
  9. (x + 5)(x + 8)
  10. 6 and 7
  11. 5 and 6
  12. 576
  13. 50
  14. 6
  15. 1444
  16. 64
  17. 4
  18. 14
  19. 56
  20. 13
  21. 1521
  22. 2 and 5
  23. 9
  24. 10404
  25. 18
  26. 8464
  27. 36
  28. 484
  29. 529
  30. 97
  31. 9025
  32. 4 and 6
  33. 81
  34. (x + 3)(x + 7)
  35. 181
  36. 185
  37. (x + 5)(x + 7)
  38. 85
  39. 1296
  40. 25

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