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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 949 · 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 48²? (a) 2704 (b) 2314 (c) 2304 (d) 2496
  2. 2.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 36) (b) (x − 4)(x − 9) (c) (x + 4)(x + 9) (d) (x + 1)(x + 12)
  3. 3.If a + b = 15 and ab = 50, what is (a − b)²? (a) 25 (b) 5 (c) 225 (d) 125
  4. 4.What is the sum of the roots of x² − 14x + 45 = 0? (a) 45 (b) 15 (c) 13 (d) 14
  5. 5.What is the sum of the roots of x² − 9x + 18 = 0? (a) 8 (b) 18 (c) 9 (d) 10
  6. 6.If a + b = 17 and ab = 60, what is a² + b²? (a) 229 (b) 289 (c) 409 (d) 169
  7. 7.If a + b = 7 and ab = 12, what is a² + b²? (a) 49 (b) 73 (c) 25 (d) 37
  8. 8.If a + b = 11 and ab = 10, what is (a − b)²? (a) 81 (b) 121 (c) 9 (d) 101
  9. 9.What is the sum of the roots of x² − 13x + 40 = 0? (a) 40 (b) 14 (c) 12 (d) 13
  10. 10.What is the product of the roots of x² − 7x + 12 = 0? (a) 13 (b) 12 (c) 11 (d) 7
  11. 11.If a + b = 14 and ab = 40, what is a² + b²? (a) 156 (b) 196 (c) 116 (d) 276
  12. 12.If a + b = 14 and ab = 40, what is (a − b)²? (a) 116 (b) 196 (c) 6 (d) 36
  13. 13.What is the product of the roots of x² − 10x + 21 = 0? (a) 21 (b) 20 (c) 22 (d) 10
  14. 14.If a + b = 17 and ab = 66, what is a² + b²? (a) 421 (b) 157 (c) 223 (d) 289
  15. 15.Which is the correct factorisation of x² + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 1)(x + 30) (c) (x + 1)(x + 10) (d) (x + 5)(x + 6)
  16. 16.If a + b = 19 and ab = 88, what is (a − b)²? (a) 9 (b) 3 (c) 185 (d) 361
  17. 17.What are the roots of x² − 15x + 54 = 0? (a) 1 and 54 (b) 2 and 27 (c) 1 and 14 (d) 6 and 9
  18. 18.If a + b = 18 and ab = 77, what is a² + b²? (a) 324 (b) 170 (c) 478 (d) 247
  19. 19.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 3025 (b) 2025 (c) 2475 (d) 2035
  20. 20.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6084 (b) 6396 (c) 6094 (d) 6724
  21. 21.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 1024 (b) 904 (c) 784 (d) 1034
  22. 22.What is the sum of the roots of x² − 15x + 56 = 0? (a) 16 (b) 14 (c) 15 (d) 56
  23. 23.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 3025 (b) 4235 (c) 4225 (d) 3625
  24. 24.If a + b = 10 and ab = 16, what is a² + b²? (a) 132 (b) 84 (c) 68 (d) 100
  25. 25.What are the roots of x² − 13x + 40 = 0? (a) 2 and 20 (b) 5 and 8 (c) 1 and 12 (d) 1 and 40
  26. 26.If a + b = 17 and ab = 60, what is (a − b)²? (a) 289 (b) 7 (c) 169 (d) 49
  27. 27.If a + b = 11 and ab = 30, what is (a − b)²? (a) 91 (b) 61 (c) 121 (d) 1
  28. 28.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 18) (b) (x − 3)(x − 6) (c) (x + 3)(x + 6) (d) (x + 1)(x + 8)
  29. 29.If a + b = 12 and ab = 32, what is (a − b)²? (a) 80 (b) 144 (c) 4 (d) 16
  30. 30.If a + b = 20 and ab = 99, what is a² + b²? (a) 202 (b) 400 (c) 301 (d) 598
  31. 31.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 4896 (b) 5184 (c) 4634 (d) 4624
  32. 32.If a + b = 10 and ab = 24, what is (a − b)²? (a) 100 (b) 2 (c) 52 (d) 4
  33. 33.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2704 (b) 2504 (c) 2304 (d) 2714
  34. 34.What is the product of the roots of x² − 15x + 56 = 0? (a) 57 (b) 56 (c) 15 (d) 55
  35. 35.Which is the correct factorisation of x² + 11x + 28? (a) (x + 4)(x + 7) (b) (x + 1)(x + 28) (c) (x + 1)(x + 10) (d) (x − 4)(x − 7)
  36. 36.If a + b = 19 and ab = 90, what is (a − b)²? (a) 361 (b) 1 (c) 181 (d) 271
  37. 37.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 4761 (b) 5041 (c) 5051 (d) 4901
  38. 38.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 884 (b) 676 (c) 1156 (d) 686
  39. 39.If a + b = 12 and ab = 11, what is (a − b)²? (a) 10 (b) 100 (c) 122 (d) 144
  40. 40.What is the product of the roots of x² − 17x + 72 = 0? (a) 72 (b) 73 (c) 71 (d) 17

Answer key

  1. 2304
  2. (x + 4)(x + 9)
  3. 25
  4. 14
  5. 9
  6. 169
  7. 25
  8. 81
  9. 13
  10. 12
  11. 116
  12. 36
  13. 21
  14. 157
  15. (x + 5)(x + 6)
  16. 9
  17. 6 and 9
  18. 170
  19. 2025
  20. 6084
  21. 1024
  22. 15
  23. 4225
  24. 68
  25. 5 and 8
  26. 49
  27. 1
  28. (x + 3)(x + 6)
  29. 16
  30. 202
  31. 4624
  32. 4
  33. 2704
  34. 56
  35. (x + 4)(x + 7)
  36. 1
  37. 5041
  38. 676
  39. 100
  40. 72

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