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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 109 · 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 + 12x + 35? (a) (x − 5)(x − 7) (b) (x + 1)(x + 11) (c) (x + 5)(x + 7) (d) (x + 1)(x + 35)
  2. 2.What is the sum of the roots of x² − 16x + 63 = 0? (a) 16 (b) 15 (c) 17 (d) 63
  3. 3.If a + b = 18 and ab = 77, what is a² + b²? (a) 478 (b) 247 (c) 324 (d) 170
  4. 4.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 4624 (b) 4904 (c) 5194 (d) 5184
  5. 5.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x − 6)(x − 8) (c) (x + 1)(x + 48) (d) (x + 6)(x + 8)
  6. 6.If a + b = 16 and ab = 48, what is a² + b²? (a) 352 (b) 208 (c) 160 (d) 256
  7. 7.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 4096 (b) 3146 (c) 3136 (d) 3584
  8. 8.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 10) (b) (x − 3)(x − 8) (c) (x + 3)(x + 8) (d) (x + 1)(x + 24)
  9. 9.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 4) (b) (x + 1)(x + 6) (c) (x + 2)(x + 3) (d) (x − 2)(x − 3)
  10. 10.If a + b = 12 and ab = 35, what is (a − b)²? (a) 2 (b) 144 (c) 4 (d) 74
  11. 11.Which is the correct factorisation of x² + 11x + 18? (a) (x + 2)(x + 9) (b) (x + 1)(x + 10) (c) (x + 1)(x + 18) (d) (x − 2)(x − 9)
  12. 12.What is the product of the roots of x² − 15x + 56 = 0? (a) 56 (b) 15 (c) 57 (d) 55
  13. 13.What is the sum of the roots of x² − 12x + 32 = 0? (a) 32 (b) 13 (c) 12 (d) 11
  14. 14.What is the product of the roots of x² − 9x + 20 = 0? (a) 20 (b) 21 (c) 9 (d) 19
  15. 15.Which is the correct factorisation of x² + 9x + 20? (a) (x + 4)(x + 5) (b) (x − 4)(x − 5) (c) (x + 1)(x + 20) (d) (x + 1)(x + 8)
  16. 16.If a + b = 10 and ab = 9, what is a² + b²? (a) 100 (b) 118 (c) 91 (d) 82
  17. 17.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)
  18. 18.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)
  19. 19.If a + b = 5 and ab = 4, what is (a − b)²? (a) 3 (b) 25 (c) 9 (d) 17
  20. 20.If a + b = 9 and ab = 20, what is (a − b)²? (a) 61 (b) 1 (c) 81 (d) 41
  21. 21.What is the sum of the roots of x² − 13x + 42 = 0? (a) 13 (b) 12 (c) 42 (d) 14
  22. 22.What is the product of the roots of x² − 16x + 63 = 0? (a) 62 (b) 64 (c) 63 (d) 16
  23. 23.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 9419 (b) 9991 (c) 10609 (d) 9409
  24. 24.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)
  25. 25.If a + b = 9 and ab = 20, what is a² + b²? (a) 61 (b) 121 (c) 41 (d) 81
  26. 26.What is the sum of the roots of x² − 14x + 48 = 0? (a) 13 (b) 48 (c) 14 (d) 15
  27. 27.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1369 (b) 1859 (c) 1849 (d) 1609
  28. 28.What is the product of the roots of x² − 13x + 36 = 0? (a) 36 (b) 37 (c) 35 (d) 13
  29. 29.If a + b = 10 and ab = 21, what is a² + b²? (a) 58 (b) 142 (c) 100 (d) 79
  30. 30.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 1024 (b) 794 (c) 784 (d) 896
  31. 31.If a + b = 13 and ab = 30, what is (a − b)²? (a) 49 (b) 169 (c) 7 (d) 109
  32. 32.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2601 (b) 2411 (c) 2499 (d) 2401
  33. 33.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 72) (c) (x + 8)(x + 9) (d) (x + 1)(x + 16)
  34. 34.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 6889 (b) 5929 (c) 6899 (d) 6409
  35. 35.What are the roots of x² − 9x + 14 = 0? (a) 2 and 7 (b) 1 and 14 (c) 1 and 8 (d) 3 and 6
  36. 36.What is the sum of the roots of x² − 11x + 18 = 0? (a) 11 (b) 18 (c) 12 (d) 10
  37. 37.What is the sum of the roots of x² − 11x + 30 = 0? (a) 30 (b) 12 (c) 10 (d) 11
  38. 38.What is the sum of the roots of x² − 7x + 10 = 0? (a) 8 (b) 6 (c) 7 (d) 10
  39. 39.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1531 (b) 1681 (c) 1599 (d) 1521
  40. 40.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 396 (b) 484 (c) 334 (d) 324

Answer key

  1. (x + 5)(x + 7)
  2. 16
  3. 170
  4. 5184
  5. (x + 6)(x + 8)
  6. 160
  7. 3136
  8. (x + 3)(x + 8)
  9. (x + 2)(x + 3)
  10. 4
  11. (x + 2)(x + 9)
  12. 56
  13. 12
  14. 20
  15. (x + 4)(x + 5)
  16. 82
  17. (x + 7)(x + 9)
  18. (x + 4)(x + 6)
  19. 9
  20. 1
  21. 13
  22. 63
  23. 9409
  24. (x + 3)(x + 6)
  25. 41
  26. 14
  27. 1849
  28. 36
  29. 58
  30. 784
  31. 49
  32. 2401
  33. (x + 8)(x + 9)
  34. 6889
  35. 2 and 7
  36. 11
  37. 11
  38. 7
  39. 1521
  40. 324

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