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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 44 · 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 = 14 and ab = 24, what is a² + b²? (a) 244 (b) 172 (c) 148 (d) 196
  2. 2.If a + b = 15 and ab = 56, what is a² + b²? (a) 113 (b) 169 (c) 337 (d) 225
  3. 3.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3844 (b) 3604 (c) 3854 (d) 3364
  4. 4.If a + b = 20 and ab = 99, what is (a − b)²? (a) 2 (b) 202 (c) 4 (d) 400
  5. 5.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 10) (b) (x − 4)(x − 7) (c) (x + 1)(x + 28) (d) (x + 4)(x + 7)
  6. 6.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 4761 (b) 5051 (c) 4901 (d) 5041
  7. 7.If a + b = 16 and ab = 60, what is a² + b²? (a) 376 (b) 196 (c) 256 (d) 136
  8. 8.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1604 (b) 1774 (c) 1764 (d) 1444
  9. 9.What are the roots of x² − 12x + 32 = 0? (a) 2 and 16 (b) 4 and 8 (c) 1 and 32 (d) 1 and 11
  10. 10.What is the product of the roots of x² − 13x + 42 = 0? (a) 42 (b) 41 (c) 43 (d) 13
  11. 11.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1099 (b) 729 (c) 909 (d) 1089
  12. 12.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 73 (c) 71 (d) 72
  13. 13.If a + b = 19 and ab = 84, what is a² + b²? (a) 193 (b) 529 (c) 277 (d) 361
  14. 14.What is the sum of the roots of x² − 8x + 15 = 0? (a) 15 (b) 9 (c) 7 (d) 8
  15. 15.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2411 (b) 2601 (c) 2499 (d) 2401
  16. 16.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4356 (b) 5476 (c) 4884 (d) 4366
  17. 17.If a + b = 12 and ab = 35, what is (a − b)²? (a) 74 (b) 2 (c) 4 (d) 144
  18. 18.If a + b = 10 and ab = 16, what is a² + b²? (a) 132 (b) 100 (c) 84 (d) 68
  19. 19.If a + b = 15 and ab = 56, what is (a − b)²? (a) 113 (b) 169 (c) 1 (d) 225
  20. 20.If a + b = 21 and ab = 108, what is a² + b²? (a) 441 (b) 657 (c) 225 (d) 333
  21. 21.If a + b = 5 and ab = 4, what is (a − b)²? (a) 25 (b) 3 (c) 17 (d) 9
  22. 22.What is the sum of the roots of x² − 16x + 63 = 0? (a) 17 (b) 15 (c) 63 (d) 16
  23. 23.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 9216 (b) 10016 (c) 10816 (d) 10826
  24. 24.If a + b = 3 and ab = 2, what is a² + b²? (a) 13 (b) 7 (c) 5 (d) 9
  25. 25.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)
  26. 26.What are the roots of x² − 12x + 27 = 0? (a) 1 and 27 (b) 1 and 11 (c) 2 and 10 (d) 3 and 9
  27. 27.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 1235 (b) 925 (c) 1225 (d) 625
  28. 28.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 56) (c) (x + 1)(x + 14) (d) (x + 7)(x + 8)
  29. 29.If a + b = 8 and ab = 12, what is (a − b)²? (a) 40 (b) 64 (c) 4 (d) 16
  30. 30.If a + b = 20 and ab = 96, what is (a − b)²? (a) 4 (b) 16 (c) 400 (d) 208
  31. 31.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x + 6)(x + 9) (c) (x + 1)(x + 14) (d) (x − 6)(x − 9)
  32. 32.What is the product of the roots of x² − 15x + 54 = 0? (a) 53 (b) 15 (c) 54 (d) 55
  33. 33.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 8) (b) (x + 2)(x + 4) (c) (x + 1)(x + 5) (d) (x − 2)(x − 4)
  34. 34.What is the product of the roots of x² − 9x + 18 = 0? (a) 18 (b) 9 (c) 19 (d) 17
  35. 35.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x + 1)(x + 15) (c) (x + 7)(x + 9) (d) (x − 7)(x − 9)
  36. 36.If a + b = 16 and ab = 63, what is a² + b²? (a) 130 (b) 193 (c) 256 (d) 382
  37. 37.What is the sum of the roots of x² − 9x + 20 = 0? (a) 9 (b) 8 (c) 20 (d) 10
  38. 38.What is the product of the roots of x² − 10x + 24 = 0? (a) 10 (b) 24 (c) 25 (d) 23
  39. 39.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)
  40. 40.Which is the correct factorisation of x² + 9x + 18? (a) (x − 3)(x − 6) (b) (x + 1)(x + 18) (c) (x + 3)(x + 6) (d) (x + 1)(x + 8)

Answer key

  1. 148
  2. 113
  3. 3844
  4. 4
  5. (x + 4)(x + 7)
  6. 5041
  7. 136
  8. 1764
  9. 4 and 8
  10. 42
  11. 1089
  12. 72
  13. 193
  14. 8
  15. 2401
  16. 4356
  17. 4
  18. 68
  19. 1
  20. 225
  21. 9
  22. 16
  23. 10816
  24. 5
  25. (x + 6)(x + 8)
  26. 3 and 9
  27. 1225
  28. (x + 7)(x + 8)
  29. 16
  30. 16
  31. (x + 6)(x + 9)
  32. 54
  33. (x + 2)(x + 4)
  34. 18
  35. (x + 7)(x + 9)
  36. 130
  37. 9
  38. 24
  39. (x + 5)(x + 6)
  40. (x + 3)(x + 6)

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