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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 428 · 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.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 739 (b) 729 (c) 891 (d) 1089
  2. 2.What is the sum of the roots of x² − 11x + 24 = 0? (a) 11 (b) 12 (c) 10 (d) 24
  3. 3.If a + b = 15 and ab = 44, what is a² + b²? (a) 181 (b) 225 (c) 313 (d) 137
  4. 4.If a + b = 7 and ab = 12, what is a² + b²? (a) 37 (b) 25 (c) 73 (d) 49
  5. 5.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)
  6. 6.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1454 (b) 1764 (c) 1444 (d) 1596
  7. 7.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 11025 (b) 9035 (c) 9975 (d) 9025
  8. 8.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x + 1)(x + 16) (c) (x + 1)(x + 72) (d) (x − 8)(x − 9)
  9. 9.What is the sum of the roots of x² − 13x + 42 = 0? (a) 12 (b) 13 (c) 42 (d) 14
  10. 10.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 6) (b) (x + 1)(x + 12) (c) (x + 3)(x + 4) (d) (x − 3)(x − 4)
  11. 11.If a + b = 11 and ab = 30, what is a² + b²? (a) 61 (b) 91 (c) 181 (d) 121
  12. 12.What are the roots of x² − 11x + 24 = 0? (a) 2 and 12 (b) 1 and 10 (c) 1 and 24 (d) 3 and 8
  13. 13.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x − 7)(x − 9) (c) (x + 1)(x + 15) (d) (x + 7)(x + 9)
  14. 14.What is the sum of the roots of x² − 12x + 27 = 0? (a) 12 (b) 27 (c) 11 (d) 13
  15. 15.What is the sum of the roots of x² − 10x + 21 = 0? (a) 11 (b) 21 (c) 9 (d) 10
  16. 16.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 10201 (b) 9801 (c) 9999 (d) 9811
  17. 17.If a + b = 17 and ab = 72, what is (a − b)²? (a) 289 (b) 145 (c) 217 (d) 1
  18. 18.What are the roots of x² − 7x + 12 = 0? (a) 3 and 4 (b) 1 and 6 (c) 1 and 12 (d) 2 and 6
  19. 19.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x − 5)(x − 7) (c) (x + 1)(x + 35) (d) (x + 5)(x + 7)
  20. 20.If a + b = 13 and ab = 36, what is (a − b)²? (a) 5 (b) 169 (c) 25 (d) 97
  21. 21.If a + b = 14 and ab = 24, what is a² + b²? (a) 196 (b) 172 (c) 148 (d) 244
  22. 22.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2209 (b) 2809 (c) 2219 (d) 2491
  23. 23.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 13) (b) (x + 5)(x + 9) (c) (x + 1)(x + 45) (d) (x − 5)(x − 9)
  24. 24.What are the roots of x² − 6x + 8 = 0? (a) 1 and 5 (b) 2 and 5 (c) 2 and 4 (d) 1 and 8
  25. 25.If a + b = 10 and ab = 24, what is a² + b²? (a) 148 (b) 76 (c) 100 (d) 52
  26. 26.If a + b = 11 and ab = 30, what is (a − b)²? (a) 61 (b) 91 (c) 121 (d) 1
  27. 27.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x − 4)(x − 5) (c) (x + 4)(x + 5) (d) (x + 1)(x + 8)
  28. 28.If a + b = 10 and ab = 16, what is a² + b²? (a) 132 (b) 100 (c) 68 (d) 84
  29. 29.What is the sum of the roots of x² − 8x + 12 = 0? (a) 12 (b) 7 (c) 9 (d) 8
  30. 30.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3854 (b) 3604 (c) 3844 (d) 3364
  31. 31.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x + 1)(x + 12) (c) (x + 6)(x + 7) (d) (x − 6)(x − 7)
  32. 32.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 904 (b) 784 (c) 1034 (d) 1024
  33. 33.If a + b = 13 and ab = 40, what is a² + b²? (a) 89 (b) 249 (c) 169 (d) 129
  34. 34.If a + b = 9 and ab = 14, what is (a − b)²? (a) 5 (b) 53 (c) 81 (d) 25
  35. 35.If a + b = 14 and ab = 40, what is a² + b²? (a) 196 (b) 276 (c) 116 (d) 156
  36. 36.If a + b = 8 and ab = 12, what is a² + b²? (a) 64 (b) 88 (c) 40 (d) 52
  37. 37.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1681 (b) 1601 (c) 1691 (d) 1521
  38. 38.If a + b = 19 and ab = 88, what is (a − b)²? (a) 361 (b) 9 (c) 185 (d) 3
  39. 39.What are the roots of x² − 5x + 6 = 0? (a) 1 and 6 (b) 2 and 3 (c) 1 and 4 (d) 2 and 4
  40. 40.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6084 (b) 6734 (c) 6404 (d) 6724

Answer key

  1. 729
  2. 11
  3. 137
  4. 25
  5. (x + 2)(x + 3)
  6. 1444
  7. 9025
  8. (x + 8)(x + 9)
  9. 13
  10. (x + 3)(x + 4)
  11. 61
  12. 3 and 8
  13. (x + 7)(x + 9)
  14. 12
  15. 10
  16. 9801
  17. 1
  18. 3 and 4
  19. (x + 5)(x + 7)
  20. 25
  21. 148
  22. 2209
  23. (x + 5)(x + 9)
  24. 2 and 4
  25. 52
  26. 1
  27. (x + 4)(x + 5)
  28. 68
  29. 8
  30. 3844
  31. (x + 6)(x + 7)
  32. 1024
  33. 89
  34. 25
  35. 116
  36. 40
  37. 1681
  38. 9
  39. 2 and 3
  40. 6724

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