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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 140 · 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 18²? (a) 324 (b) 484 (c) 396 (d) 334
  2. 2.What is the sum of the roots of x² − 10x + 21 = 0? (a) 10 (b) 11 (c) 9 (d) 21
  3. 3.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 404 (b) 484 (c) 494 (d) 324
  4. 4.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x − 5)(x − 6) (c) (x + 1)(x + 10) (d) (x + 5)(x + 6)
  5. 5.What is the product of the roots of x² − 14x + 45 = 0? (a) 44 (b) 45 (c) 14 (d) 46
  6. 6.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)
  7. 7.If a + b = 12 and ab = 11, what is (a − b)²? (a) 144 (b) 122 (c) 10 (d) 100
  8. 8.If a + b = 5 and ab = 4, what is a² + b²? (a) 21 (b) 17 (c) 25 (d) 33
  9. 9.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x − 5)(x − 7) (c) (x + 5)(x + 7) (d) (x + 1)(x + 35)
  10. 10.If a + b = 12 and ab = 20, what is a² + b²? (a) 184 (b) 104 (c) 124 (d) 144
  11. 11.What is the product of the roots of x² − 12x + 35 = 0? (a) 12 (b) 35 (c) 34 (d) 36
  12. 12.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 15) (c) (x + 7)(x + 9) (d) (x + 1)(x + 63)
  13. 13.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 7235 (b) 8075 (c) 9025 (d) 7225
  14. 14.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 5051 (b) 4901 (c) 5041 (d) 4761
  15. 15.What are the roots of x² − 12x + 27 = 0? (a) 1 and 27 (b) 3 and 9 (c) 1 and 11 (d) 2 and 10
  16. 16.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 17 (c) 16 (d) 18
  17. 17.If a + b = 19 and ab = 90, what is (a − b)²? (a) 361 (b) 181 (c) 1 (d) 271
  18. 18.What are the roots of x² − 12x + 32 = 0? (a) 1 and 32 (b) 1 and 11 (c) 2 and 16 (d) 4 and 8
  19. 19.What is the product of the roots of x² − 13x + 40 = 0? (a) 13 (b) 39 (c) 40 (d) 41
  20. 20.If a + b = 15 and ab = 56, what is a² + b²? (a) 113 (b) 225 (c) 169 (d) 337
  21. 21.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 14) (b) (x + 1)(x + 54) (c) (x + 6)(x + 9) (d) (x − 6)(x − 9)
  22. 22.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1379 (b) 1369 (c) 1591 (d) 1849
  23. 23.If a + b = 12 and ab = 27, what is a² + b²? (a) 198 (b) 90 (c) 117 (d) 144
  24. 24.What are the roots of x² − 8x + 15 = 0? (a) 1 and 15 (b) 1 and 7 (c) 2 and 6 (d) 3 and 5
  25. 25.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 529 (b) 289 (c) 299 (d) 391
  26. 26.If a + b = 12 and ab = 20, what is (a − b)²? (a) 64 (b) 144 (c) 8 (d) 104
  27. 27.What are the roots of x² − 16x + 63 = 0? (a) 7 and 9 (b) 1 and 63 (c) 3 and 21 (d) 1 and 15
  28. 28.If a + b = 20 and ab = 99, what is a² + b²? (a) 202 (b) 301 (c) 400 (d) 598
  29. 29.If a + b = 19 and ab = 84, what is (a − b)²? (a) 193 (b) 25 (c) 5 (d) 361
  30. 30.If a + b = 13 and ab = 42, what is a² + b²? (a) 85 (b) 253 (c) 127 (d) 169
  31. 31.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)
  32. 32.If a + b = 13 and ab = 36, what is a² + b²? (a) 241 (b) 169 (c) 133 (d) 97
  33. 33.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1444 (b) 1454 (c) 1596 (d) 1764
  34. 34.What are the roots of x² − 9x + 14 = 0? (a) 3 and 6 (b) 1 and 8 (c) 1 and 14 (d) 2 and 7
  35. 35.If a + b = 17 and ab = 70, what is a² + b²? (a) 429 (b) 289 (c) 149 (d) 219
  36. 36.If a + b = 11 and ab = 18, what is a² + b²? (a) 121 (b) 103 (c) 157 (d) 85
  37. 37.What is the sum of the roots of x² − 13x + 42 = 0? (a) 13 (b) 14 (c) 12 (d) 42
  38. 38.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1681 (b) 1531 (c) 1521 (d) 1599
  39. 39.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 4096 (b) 3136 (c) 3584 (d) 3146
  40. 40.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 9025 (b) 11025 (c) 10025 (d) 11035

Answer key

  1. 324
  2. 10
  3. 484
  4. (x + 5)(x + 6)
  5. 45
  6. (x + 6)(x + 7)
  7. 100
  8. 17
  9. (x + 5)(x + 7)
  10. 104
  11. 35
  12. (x + 7)(x + 9)
  13. 7225
  14. 5041
  15. 3 and 9
  16. 17
  17. 1
  18. 4 and 8
  19. 40
  20. 113
  21. (x + 6)(x + 9)
  22. 1369
  23. 90
  24. 3 and 5
  25. 289
  26. 64
  27. 7 and 9
  28. 202
  29. 25
  30. 85
  31. (x + 8)(x + 9)
  32. 97
  33. 1444
  34. 2 and 7
  35. 149
  36. 85
  37. 13
  38. 1521
  39. 3136
  40. 11025

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