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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 740 · 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 + 15x + 54? (a) (x + 6)(x + 9) (b) (x + 1)(x + 14) (c) (x + 1)(x + 54) (d) (x − 6)(x − 9)
  2. 2.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6561 (b) 6571 (c) 6401 (d) 6241
  3. 3.If a + b = 15 and ab = 44, what is a² + b²? (a) 181 (b) 313 (c) 225 (d) 137
  4. 4.If a + b = 10 and ab = 21, what is (a − b)²? (a) 4 (b) 100 (c) 58 (d) 16
  5. 5.If a + b = 16 and ab = 60, what is a² + b²? (a) 136 (b) 196 (c) 256 (d) 376
  6. 6.If a + b = 13 and ab = 22, what is (a − b)²? (a) 125 (b) 169 (c) 81 (d) 9
  7. 7.What is the sum of the roots of x² − 12x + 35 = 0? (a) 11 (b) 13 (c) 12 (d) 35
  8. 8.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 45) (b) (x + 1)(x + 13) (c) (x + 5)(x + 9) (d) (x − 5)(x − 9)
  9. 9.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1521 (b) 1691 (c) 1601 (d) 1681
  10. 10.If a + b = 4 and ab = 3, what is (a − b)²? (a) 2 (b) 16 (c) 4 (d) 10
  11. 11.What is the product of the roots of x² − 11x + 18 = 0? (a) 19 (b) 17 (c) 18 (d) 11
  12. 12.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x + 1)(x + 35) (c) (x − 5)(x − 7) (d) (x + 5)(x + 7)
  13. 13.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3969 (b) 3979 (c) 3249 (d) 3609
  14. 14.What are the roots of x² − 13x + 42 = 0? (a) 2 and 21 (b) 1 and 42 (c) 1 and 12 (d) 6 and 7
  15. 15.What are the roots of x² − 13x + 36 = 0? (a) 1 and 12 (b) 2 and 18 (c) 1 and 36 (d) 4 and 9
  16. 16.If a + b = 10 and ab = 9, what is a² + b²? (a) 118 (b) 91 (c) 82 (d) 100
  17. 17.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2714 (b) 2304 (c) 2704 (d) 2504
  18. 18.Which is the correct factorisation of x² + 10x + 21? (a) (x − 3)(x − 7) (b) (x + 1)(x + 21) (c) (x + 3)(x + 7) (d) (x + 1)(x + 9)
  19. 19.What is the sum of the roots of x² − 15x + 54 = 0? (a) 15 (b) 16 (c) 54 (d) 14
  20. 20.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 18 (c) 72 (d) 17
  21. 21.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3844 (b) 3374 (c) 3364 (d) 3596
  22. 22.If a + b = 9 and ab = 20, what is (a − b)²? (a) 81 (b) 41 (c) 61 (d) 1
  23. 23.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 10) (b) (x + 2)(x + 9) (c) (x − 2)(x − 9) (d) (x + 1)(x + 18)
  24. 24.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6241 (b) 6561 (c) 6399 (d) 6251
  25. 25.If a + b = 6 and ab = 8, what is (a − b)²? (a) 36 (b) 20 (c) 2 (d) 4
  26. 26.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 9035 (b) 8125 (c) 7225 (d) 9025
  27. 27.What is the sum of the roots of x² − 15x + 56 = 0? (a) 16 (b) 56 (c) 15 (d) 14
  28. 28.Which is the correct factorisation of x² + 12x + 27? (a) (x + 3)(x + 9) (b) (x + 1)(x + 27) (c) (x − 3)(x − 9) (d) (x + 1)(x + 11)
  29. 29.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x + 7)(x + 8) (c) (x + 1)(x + 56) (d) (x − 7)(x − 8)
  30. 30.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 625 (b) 1225 (c) 875 (d) 635
  31. 31.What is the product of the roots of x² − 5x + 6 = 0? (a) 8 (b) 5 (c) 6 (d) 7
  32. 32.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 12) (b) (x − 2)(x − 6) (c) (x + 2)(x + 6) (d) (x + 1)(x + 7)
  33. 33.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9035 (b) 11025 (c) 9975 (d) 9025
  34. 34.What are the roots of x² − 5x + 6 = 0? (a) 1 and 4 (b) 2 and 4 (c) 1 and 6 (d) 2 and 3
  35. 35.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 4916 (b) 4356 (c) 5486 (d) 5476
  36. 36.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 1156 (b) 1166 (c) 676 (d) 916
  37. 37.If a + b = 13 and ab = 30, what is a² + b²? (a) 229 (b) 109 (c) 169 (d) 139
  38. 38.If a + b = 16 and ab = 48, what is (a − b)²? (a) 64 (b) 256 (c) 160 (d) 8
  39. 39.If a + b = 17 and ab = 60, what is (a − b)²? (a) 49 (b) 289 (c) 7 (d) 169
  40. 40.If a + b = 14 and ab = 33, what is (a − b)²? (a) 196 (b) 8 (c) 130 (d) 64

Answer key

  1. (x + 6)(x + 9)
  2. 6561
  3. 137
  4. 16
  5. 136
  6. 81
  7. 12
  8. (x + 5)(x + 9)
  9. 1681
  10. 4
  11. 18
  12. (x + 5)(x + 7)
  13. 3969
  14. 6 and 7
  15. 4 and 9
  16. 82
  17. 2704
  18. (x + 3)(x + 7)
  19. 15
  20. 17
  21. 3364
  22. 1
  23. (x + 2)(x + 9)
  24. 6241
  25. 4
  26. 9025
  27. 15
  28. (x + 3)(x + 9)
  29. (x + 7)(x + 8)
  30. 625
  31. 6
  32. (x + 2)(x + 6)
  33. 9025
  34. 2 and 3
  35. 5476
  36. 1156
  37. 109
  38. 64
  39. 49
  40. 64

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