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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 526 · 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 = 13 and ab = 42, what is a² + b²? (a) 169 (b) 127 (c) 253 (d) 85
  2. 2.What is the sum of the roots of x² − 8x + 12 = 0? (a) 8 (b) 12 (c) 7 (d) 9
  3. 3.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 8) (b) (x + 1)(x + 18) (c) (x + 3)(x + 6) (d) (x − 3)(x − 6)
  4. 4.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x + 6)(x + 8) (c) (x − 6)(x − 8) (d) (x + 1)(x + 13)
  5. 5.If a + b = 17 and ab = 66, what is a² + b²? (a) 289 (b) 223 (c) 157 (d) 421
  6. 6.If a + b = 21 and ab = 108, what is (a − b)²? (a) 9 (b) 441 (c) 3 (d) 225
  7. 7.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 901 (b) 841 (c) 971 (d) 961
  8. 8.What is the sum of the roots of x² − 8x + 15 = 0? (a) 8 (b) 7 (c) 9 (d) 15
  9. 9.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 5929 (b) 6409 (c) 6889 (d) 6899
  10. 10.What is the sum of the roots of x² − 15x + 56 = 0? (a) 16 (b) 15 (c) 14 (d) 56
  11. 11.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 1034 (b) 784 (c) 904 (d) 1024
  12. 12.What is the sum of the roots of x² − 7x + 10 = 0? (a) 8 (b) 7 (c) 6 (d) 10
  13. 13.If a + b = 13 and ab = 40, what is (a − b)²? (a) 3 (b) 169 (c) 89 (d) 9
  14. 14.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 4225 (b) 4235 (c) 5625 (d) 4875
  15. 15.If a + b = 14 and ab = 33, what is a² + b²? (a) 163 (b) 262 (c) 196 (d) 130
  16. 16.What are the roots of x² − 11x + 24 = 0? (a) 2 and 12 (b) 1 and 10 (c) 3 and 8 (d) 1 and 24
  17. 17.If a + b = 20 and ab = 96, what is (a − b)²? (a) 208 (b) 4 (c) 400 (d) 16
  18. 18.If a + b = 10 and ab = 16, what is (a − b)²? (a) 6 (b) 68 (c) 36 (d) 100
  19. 19.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 7579 (b) 8649 (c) 8091 (d) 7569
  20. 20.If a + b = 14 and ab = 45, what is (a − b)²? (a) 16 (b) 4 (c) 106 (d) 196
  21. 21.If a + b = 10 and ab = 24, what is (a − b)²? (a) 4 (b) 2 (c) 100 (d) 52
  22. 22.Which is the correct factorisation of x² + 9x + 20? (a) (x + 4)(x + 5) (b) (x − 4)(x − 5) (c) (x + 1)(x + 20) (d) (x + 1)(x + 8)
  23. 23.What is the sum of the roots of x² − 7x + 12 = 0? (a) 12 (b) 8 (c) 6 (d) 7
  24. 24.If a + b = 4 and ab = 3, what is a² + b²? (a) 10 (b) 16 (c) 22 (d) 13
  25. 25.If a + b = 7 and ab = 12, what is (a − b)²? (a) 25 (b) 37 (c) 1 (d) 49
  26. 26.If a + b = 20 and ab = 96, what is a² + b²? (a) 400 (b) 592 (c) 304 (d) 208
  27. 27.If a + b = 19 and ab = 84, what is a² + b²? (a) 193 (b) 361 (c) 277 (d) 529
  28. 28.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 1225 (b) 2035 (c) 1625 (d) 2025
  29. 29.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 625 (b) 635 (c) 1225 (d) 875
  30. 30.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 7235 (b) 6425 (c) 7225 (d) 5625
  31. 31.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x − 8)(x − 9) (c) (x + 8)(x + 9) (d) (x + 1)(x + 72)
  32. 32.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8474 (b) 7744 (c) 8464 (d) 8104
  33. 33.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 17 (c) 16 (d) 18
  34. 34.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x − 7)(x − 9) (c) (x + 1)(x + 63) (d) (x + 7)(x + 9)
  35. 35.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 266 (b) 576 (c) 256 (d) 384
  36. 36.What is the sum of the roots of x² − 16x + 63 = 0? (a) 17 (b) 63 (c) 15 (d) 16
  37. 37.What is the sum of the roots of x² − 6x + 8 = 0? (a) 6 (b) 7 (c) 8 (d) 5
  38. 38.If a + b = 8 and ab = 15, what is (a − b)²? (a) 34 (b) 2 (c) 4 (d) 64
  39. 39.What are the roots of x² − 9x + 18 = 0? (a) 1 and 18 (b) 1 and 8 (c) 3 and 6 (d) 2 and 9
  40. 40.What are the roots of x² − 10x + 16 = 0? (a) 1 and 9 (b) 2 and 8 (c) 3 and 7 (d) 1 and 16

Answer key

  1. 85
  2. 8
  3. (x + 3)(x + 6)
  4. (x + 6)(x + 8)
  5. 157
  6. 9
  7. 961
  8. 8
  9. 6889
  10. 15
  11. 1024
  12. 7
  13. 9
  14. 4225
  15. 130
  16. 3 and 8
  17. 16
  18. 36
  19. 7569
  20. 16
  21. 4
  22. (x + 4)(x + 5)
  23. 7
  24. 10
  25. 1
  26. 208
  27. 193
  28. 2025
  29. 625
  30. 7225
  31. (x + 8)(x + 9)
  32. 8464
  33. 17
  34. (x + 7)(x + 9)
  35. 256
  36. 16
  37. 6
  38. 4
  39. 3 and 6
  40. 2 and 8

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