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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 963 · 40 questions
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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 74²? (a) 4356 (b) 5476 (c) 5486 (d) 4916
  2. 2.If a + b = 14 and ab = 48, what is a² + b²? (a) 292 (b) 148 (c) 100 (d) 196
  3. 3.Which is the correct factorisation of x² + 13x + 36? (a) (x + 4)(x + 9) (b) (x + 1)(x + 36) (c) (x − 4)(x − 9) (d) (x + 1)(x + 12)
  4. 4.If a + b = 17 and ab = 72, what is a² + b²? (a) 433 (b) 289 (c) 145 (d) 217
  5. 5.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 5) (b) (x + 2)(x + 4) (c) (x − 2)(x − 4) (d) (x + 1)(x + 8)
  6. 6.What is the product of the roots of x² − 9x + 18 = 0? (a) 18 (b) 17 (c) 9 (d) 19
  7. 7.If a + b = 8 and ab = 15, what is (a − b)²? (a) 34 (b) 2 (c) 64 (d) 4
  8. 8.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8281 (b) 8101 (c) 8291 (d) 7921
  9. 9.Which is the correct factorisation of x² + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 5)(x + 6) (c) (x + 1)(x + 30) (d) (x + 1)(x + 10)
  10. 10.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3844 (b) 3374 (c) 3364 (d) 3596
  11. 11.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 6409 (b) 6889 (c) 5929 (d) 6899
  12. 12.If a + b = 17 and ab = 60, what is a² + b²? (a) 409 (b) 289 (c) 229 (d) 169
  13. 13.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 73 (c) 17 (d) 72
  14. 14.What is the product of the roots of x² − 13x + 36 = 0? (a) 13 (b) 35 (c) 37 (d) 36
  15. 15.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 21) (b) (x + 1)(x + 9) (c) (x + 3)(x + 7) (d) (x − 3)(x − 7)
  16. 16.If a + b = 4 and ab = 3, what is (a − b)²? (a) 2 (b) 16 (c) 4 (d) 10
  17. 17.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2809 (b) 2209 (c) 2219 (d) 2491
  18. 18.If a + b = 13 and ab = 42, what is (a − b)²? (a) 85 (b) 1 (c) 169 (d) 127
  19. 19.If a + b = 13 and ab = 40, what is a² + b²? (a) 89 (b) 169 (c) 129 (d) 249
  20. 20.What is the product of the roots of x² − 11x + 30 = 0? (a) 31 (b) 11 (c) 29 (d) 30
  21. 21.Which is the correct factorisation of x² + 12x + 32? (a) (x + 4)(x + 8) (b) (x + 1)(x + 32) (c) (x + 1)(x + 11) (d) (x − 4)(x − 8)
  22. 22.Which is the correct factorisation of x² + 13x + 42? (a) (x + 6)(x + 7) (b) (x + 1)(x + 42) (c) (x − 6)(x − 7) (d) (x + 1)(x + 12)
  23. 23.What is the product of the roots of x² − 14x + 45 = 0? (a) 46 (b) 44 (c) 14 (d) 45
  24. 24.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 35) (b) (x − 5)(x − 7) (c) (x + 5)(x + 7) (d) (x + 1)(x + 11)
  25. 25.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 884 (b) 676 (c) 686 (d) 1156
  26. 26.If a + b = 18 and ab = 80, what is (a − b)²? (a) 324 (b) 2 (c) 4 (d) 164
  27. 27.If a + b = 9 and ab = 8, what is a² + b²? (a) 81 (b) 65 (c) 97 (d) 73
  28. 28.If a + b = 14 and ab = 40, what is (a − b)²? (a) 36 (b) 196 (c) 116 (d) 6
  29. 29.If a + b = 8 and ab = 7, what is (a − b)²? (a) 50 (b) 36 (c) 6 (d) 64
  30. 30.What is the sum of the roots of x² − 14x + 48 = 0? (a) 48 (b) 13 (c) 14 (d) 15
  31. 31.If a + b = 13 and ab = 36, what is a² + b²? (a) 97 (b) 169 (c) 133 (d) 241
  32. 32.If a + b = 7 and ab = 12, what is (a − b)²? (a) 37 (b) 1 (c) 25 (d) 49
  33. 33.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 16 (c) 17 (d) 72
  34. 34.If a + b = 17 and ab = 66, what is (a − b)²? (a) 157 (b) 25 (c) 289 (d) 5
  35. 35.What is the sum of the roots of x² − 7x + 12 = 0? (a) 7 (b) 6 (c) 8 (d) 12
  36. 36.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9975 (b) 11025 (c) 9035 (d) 9025
  37. 37.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 266 (b) 576 (c) 256 (d) 384
  38. 38.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 6384 (b) 5786 (c) 7056 (d) 5776
  39. 39.If a + b = 10 and ab = 16, what is (a − b)²? (a) 6 (b) 68 (c) 100 (d) 36
  40. 40.If a + b = 15 and ab = 50, what is a² + b²? (a) 225 (b) 325 (c) 125 (d) 175

Answer key

  1. 5476
  2. 100
  3. (x + 4)(x + 9)
  4. 145
  5. (x + 2)(x + 4)
  6. 18
  7. 4
  8. 8281
  9. (x + 5)(x + 6)
  10. 3364
  11. 6889
  12. 169
  13. 72
  14. 36
  15. (x + 3)(x + 7)
  16. 4
  17. 2209
  18. 1
  19. 89
  20. 30
  21. (x + 4)(x + 8)
  22. (x + 6)(x + 7)
  23. 45
  24. (x + 5)(x + 7)
  25. 676
  26. 4
  27. 65
  28. 36
  29. 36
  30. 14
  31. 97
  32. 1
  33. 17
  34. 25
  35. 7
  36. 9025
  37. 256
  38. 5776
  39. 36
  40. 125

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