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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 101 · 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 (a − b)²? (a) 400 (b) 2 (c) 4 (d) 202
  2. 2.If a + b = 18 and ab = 77, what is (a − b)²? (a) 324 (b) 4 (c) 170 (d) 16
  3. 3.If a + b = 15 and ab = 44, what is (a − b)²? (a) 49 (b) 7 (c) 225 (d) 137
  4. 4.If a + b = 17 and ab = 70, what is a² + b²? (a) 289 (b) 429 (c) 219 (d) 149
  5. 5.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 16 (c) 72 (d) 17
  6. 6.If a + b = 12 and ab = 35, what is (a − b)²? (a) 144 (b) 74 (c) 2 (d) 4
  7. 7.If a + b = 7 and ab = 10, what is a² + b²? (a) 29 (b) 49 (c) 39 (d) 69
  8. 8.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 73 (c) 71 (d) 72
  9. 9.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 1)(x + 72) (c) (x − 8)(x − 9) (d) (x + 8)(x + 9)
  10. 10.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 6) (b) (x − 2)(x − 3) (c) (x + 1)(x + 4) (d) (x + 2)(x + 3)
  11. 11.If a + b = 15 and ab = 56, what is a² + b²? (a) 113 (b) 225 (c) 337 (d) 169
  12. 12.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8101 (b) 7921 (c) 8291 (d) 8281
  13. 13.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 784 (b) 1034 (c) 904 (d) 1024
  14. 14.If a + b = 16 and ab = 63, what is (a − b)²? (a) 256 (b) 2 (c) 4 (d) 130
  15. 15.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 6899 (b) 6409 (c) 5929 (d) 6889
  16. 16.If a + b = 19 and ab = 90, what is a² + b²? (a) 361 (b) 541 (c) 181 (d) 271
  17. 17.If a + b = 7 and ab = 6, what is a² + b²? (a) 37 (b) 61 (c) 49 (d) 43
  18. 18.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 971 (b) 961 (c) 841 (d) 901
  19. 19.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 729 (b) 1089 (c) 909 (d) 1099
  20. 20.If a + b = 18 and ab = 72, what is a² + b²? (a) 252 (b) 324 (c) 180 (d) 468
  21. 21.What is the sum of the roots of x² − 15x + 56 = 0? (a) 16 (b) 14 (c) 56 (d) 15
  22. 22.What are the roots of x² − 12x + 32 = 0? (a) 2 and 16 (b) 1 and 11 (c) 1 and 32 (d) 4 and 8
  23. 23.If a + b = 19 and ab = 88, what is a² + b²? (a) 361 (b) 273 (c) 185 (d) 537
  24. 24.What are the roots of x² − 13x + 40 = 0? (a) 1 and 12 (b) 2 and 20 (c) 1 and 40 (d) 5 and 8
  25. 25.If a + b = 14 and ab = 45, what is a² + b²? (a) 286 (b) 196 (c) 106 (d) 151
  26. 26.What are the roots of x² − 16x + 63 = 0? (a) 7 and 9 (b) 1 and 63 (c) 1 and 15 (d) 3 and 21
  27. 27.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 3035 (b) 3025 (c) 3575 (d) 4225
  28. 28.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3969 (b) 3249 (c) 3591 (d) 3259
  29. 29.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10201 (b) 10211 (c) 10001 (d) 9801
  30. 30.If a + b = 13 and ab = 40, what is a² + b²? (a) 129 (b) 89 (c) 249 (d) 169
  31. 31.Which is the correct factorisation of x² + 8x + 15? (a) (x − 3)(x − 5) (b) (x + 1)(x + 15) (c) (x + 3)(x + 5) (d) (x + 1)(x + 7)
  32. 32.Which is the correct factorisation of x² + 9x + 14? (a) (x + 2)(x + 7) (b) (x − 2)(x − 7) (c) (x + 1)(x + 14) (d) (x + 1)(x + 8)
  33. 33.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 6) (b) (x + 2)(x + 5) (c) (x + 1)(x + 10) (d) (x − 2)(x − 5)
  34. 34.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)
  35. 35.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 9604 (b) 10414 (c) 10004 (d) 10404
  36. 36.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 12) (b) (x + 1)(x + 36) (c) (x − 4)(x − 9) (d) (x + 4)(x + 9)
  37. 37.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)
  38. 38.What are the roots of x² − 9x + 14 = 0? (a) 3 and 6 (b) 1 and 8 (c) 2 and 7 (d) 1 and 14
  39. 39.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 9216 (b) 10016 (c) 10826 (d) 10816
  40. 40.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 9) (b) (x + 3)(x + 7) (c) (x + 1)(x + 21) (d) (x − 3)(x − 7)

Answer key

  1. 4
  2. 16
  3. 49
  4. 149
  5. 17
  6. 4
  7. 29
  8. 72
  9. (x + 8)(x + 9)
  10. (x + 2)(x + 3)
  11. 113
  12. 8281
  13. 1024
  14. 4
  15. 6889
  16. 181
  17. 37
  18. 961
  19. 1089
  20. 180
  21. 15
  22. 4 and 8
  23. 185
  24. 5 and 8
  25. 106
  26. 7 and 9
  27. 3025
  28. 3249
  29. 10201
  30. 89
  31. (x + 3)(x + 5)
  32. (x + 2)(x + 7)
  33. (x + 2)(x + 5)
  34. (x + 6)(x + 8)
  35. 10404
  36. (x + 4)(x + 9)
  37. (x + 5)(x + 7)
  38. 2 and 7
  39. 10816
  40. (x + 3)(x + 7)

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