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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 585 · 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 = 20 and ab = 99, what is a² + b²? (a) 598 (b) 400 (c) 202 (d) 301
  2. 2.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 6889 (b) 5929 (c) 6409 (d) 6899
  3. 3.What are the roots of x² − 16x + 63 = 0? (a) 1 and 63 (b) 3 and 21 (c) 7 and 9 (d) 1 and 15
  4. 4.What is the product of the roots of x² − 15x + 56 = 0? (a) 56 (b) 57 (c) 55 (d) 15
  5. 5.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4771 (b) 4761 (c) 5041 (d) 4899
  6. 6.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x − 5)(x − 6) (c) (x + 5)(x + 6) (d) (x + 1)(x + 10)
  7. 7.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 916 (b) 1156 (c) 1166 (d) 676
  8. 8.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 14) (c) (x + 1)(x + 56) (d) (x + 7)(x + 8)
  9. 9.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 8116 (b) 8846 (c) 8836 (d) 7396
  10. 10.Which is the correct factorisation of x² + 7x + 10? (a) (x − 2)(x − 5) (b) (x + 1)(x + 6) (c) (x + 2)(x + 5) (d) (x + 1)(x + 10)
  11. 11.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 6)(x + 7) (c) (x + 1)(x + 42) (d) (x + 1)(x + 12)
  12. 12.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 24) (b) (x − 3)(x − 8) (c) (x + 3)(x + 8) (d) (x + 1)(x + 10)
  13. 13.Which is the correct factorisation of x² + 6x + 8? (a) (x − 2)(x − 4) (b) (x + 1)(x + 8) (c) (x + 1)(x + 5) (d) (x + 2)(x + 4)
  14. 14.What are the roots of x² − 13x + 42 = 0? (a) 2 and 21 (b) 6 and 7 (c) 1 and 12 (d) 1 and 42
  15. 15.If a + b = 13 and ab = 40, what is a² + b²? (a) 129 (b) 169 (c) 249 (d) 89
  16. 16.If a + b = 8 and ab = 15, what is a² + b²? (a) 94 (b) 64 (c) 49 (d) 34
  17. 17.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 784 (b) 904 (c) 1034 (d) 1024
  18. 18.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 686 (b) 676 (c) 1156 (d) 884
  19. 19.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8101 (b) 7921 (c) 8291 (d) 8281
  20. 20.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2209 (b) 2819 (c) 2509 (d) 2809
  21. 21.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 1)(x + 48) (c) (x + 1)(x + 13) (d) (x + 6)(x + 8)
  22. 22.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 4916 (b) 4356 (c) 5486 (d) 5476
  23. 23.What is the product of the roots of x² − 10x + 21 = 0? (a) 22 (b) 20 (c) 10 (d) 21
  24. 24.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 16 (c) 17 (d) 18
  25. 25.If a + b = 10 and ab = 21, what is (a − b)²? (a) 58 (b) 4 (c) 100 (d) 16
  26. 26.If a + b = 15 and ab = 44, what is (a − b)²? (a) 7 (b) 225 (c) 49 (d) 137
  27. 27.What is the sum of the roots of x² − 9x + 14 = 0? (a) 9 (b) 14 (c) 8 (d) 10
  28. 28.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1946 (b) 1936 (c) 1616 (d) 1296
  29. 29.If a + b = 9 and ab = 18, what is a² + b²? (a) 81 (b) 63 (c) 117 (d) 45
  30. 30.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6724 (b) 6084 (c) 6734 (d) 6404
  31. 31.What is the product of the roots of x² − 14x + 48 = 0? (a) 14 (b) 49 (c) 48 (d) 47
  32. 32.If a + b = 14 and ab = 48, what is a² + b²? (a) 292 (b) 196 (c) 100 (d) 148
  33. 33.What is the sum of the roots of x² − 13x + 40 = 0? (a) 40 (b) 13 (c) 14 (d) 12
  34. 34.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6399 (b) 6561 (c) 6241 (d) 6251
  35. 35.If a + b = 17 and ab = 70, what is a² + b²? (a) 149 (b) 219 (c) 289 (d) 429
  36. 36.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 10) (b) (x + 1)(x + 18) (c) (x + 2)(x + 9) (d) (x − 2)(x − 9)
  37. 37.Which is the correct factorisation of x² + 8x + 15? (a) (x + 3)(x + 5) (b) (x + 1)(x + 7) (c) (x + 1)(x + 15) (d) (x − 3)(x − 5)
  38. 38.What are the roots of x² − 9x + 18 = 0? (a) 3 and 6 (b) 2 and 9 (c) 1 and 8 (d) 1 and 18
  39. 39.What are the roots of x² − 15x + 56 = 0? (a) 7 and 8 (b) 2 and 28 (c) 1 and 14 (d) 1 and 56
  40. 40.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 17 (c) 73 (d) 72

Answer key

  1. 202
  2. 6889
  3. 7 and 9
  4. 56
  5. 4761
  6. (x + 5)(x + 6)
  7. 1156
  8. (x + 7)(x + 8)
  9. 8836
  10. (x + 2)(x + 5)
  11. (x + 6)(x + 7)
  12. (x + 3)(x + 8)
  13. (x + 2)(x + 4)
  14. 6 and 7
  15. 89
  16. 34
  17. 1024
  18. 676
  19. 8281
  20. 2809
  21. (x + 6)(x + 8)
  22. 5476
  23. 21
  24. 17
  25. 16
  26. 49
  27. 9
  28. 1936
  29. 45
  30. 6724
  31. 48
  32. 100
  33. 13
  34. 6241
  35. 149
  36. (x + 2)(x + 9)
  37. (x + 3)(x + 5)
  38. 3 and 6
  39. 7 and 8
  40. 72

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