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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 92 · 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 = 16 and ab = 48, what is a² + b²? (a) 160 (b) 352 (c) 256 (d) 208
  2. 2.What are the roots of x² − 13x + 42 = 0? (a) 1 and 12 (b) 2 and 21 (c) 1 and 42 (d) 6 and 7
  3. 3.Which is the correct factorisation of x² + 8x + 15? (a) (x + 3)(x + 5) (b) (x − 3)(x − 5) (c) (x + 1)(x + 7) (d) (x + 1)(x + 15)
  4. 4.Which is the correct factorisation of x² + 9x + 18? (a) (x − 3)(x − 6) (b) (x + 1)(x + 18) (c) (x + 3)(x + 6) (d) (x + 1)(x + 8)
  5. 5.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 35) (b) (x + 1)(x + 11) (c) (x − 5)(x − 7) (d) (x + 5)(x + 7)
  6. 6.If a + b = 14 and ab = 48, what is a² + b²? (a) 196 (b) 148 (c) 100 (d) 292
  7. 7.What are the roots of x² − 13x + 40 = 0? (a) 1 and 12 (b) 1 and 40 (c) 5 and 8 (d) 2 and 20
  8. 8.If a + b = 19 and ab = 90, what is (a − b)²? (a) 181 (b) 271 (c) 1 (d) 361
  9. 9.If a + b = 11 and ab = 30, what is (a − b)²? (a) 1 (b) 121 (c) 61 (d) 91
  10. 10.Which is the correct factorisation of x² + 8x + 12? (a) (x − 2)(x − 6) (b) (x + 1)(x + 7) (c) (x + 2)(x + 6) (d) (x + 1)(x + 12)
  11. 11.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 6) (b) (x − 2)(x − 3) (c) (x + 2)(x + 3) (d) (x + 1)(x + 4)
  12. 12.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 16) (b) (x − 2)(x − 8) (c) (x + 1)(x + 9) (d) (x + 2)(x + 8)
  13. 13.If a + b = 5 and ab = 6, what is a² + b²? (a) 19 (b) 25 (c) 37 (d) 13
  14. 14.If a + b = 18 and ab = 72, what is (a − b)²? (a) 180 (b) 324 (c) 6 (d) 36
  15. 15.What are the roots of x² − 15x + 56 = 0? (a) 7 and 8 (b) 1 and 14 (c) 1 and 56 (d) 2 and 28
  16. 16.If a + b = 10 and ab = 21, what is (a − b)²? (a) 16 (b) 58 (c) 100 (d) 4
  17. 17.What is the product of the roots of x² − 7x + 12 = 0? (a) 11 (b) 13 (c) 7 (d) 12
  18. 18.What are the roots of x² − 15x + 54 = 0? (a) 1 and 54 (b) 2 and 27 (c) 6 and 9 (d) 1 and 14
  19. 19.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 7754 (b) 8096 (c) 7744 (d) 8464
  20. 20.What is the sum of the roots of x² − 15x + 54 = 0? (a) 15 (b) 54 (c) 14 (d) 16
  21. 21.If a + b = 3 and ab = 2, what is (a − b)²? (a) 9 (b) 5 (c) 7 (d) 1
  22. 22.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 7066 (b) 5776 (c) 6416 (d) 7056
  23. 23.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1774 (b) 1604 (c) 1444 (d) 1764
  24. 24.If a + b = 16 and ab = 55, what is (a − b)²? (a) 256 (b) 36 (c) 146 (d) 6
  25. 25.If a + b = 10 and ab = 24, what is a² + b²? (a) 52 (b) 148 (c) 100 (d) 76
  26. 26.If a + b = 13 and ab = 22, what is (a − b)²? (a) 169 (b) 9 (c) 125 (d) 81
  27. 27.If a + b = 14 and ab = 33, what is a² + b²? (a) 163 (b) 262 (c) 196 (d) 130
  28. 28.What is the sum of the roots of x² − 9x + 20 = 0? (a) 20 (b) 9 (c) 8 (d) 10
  29. 29.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 18 (c) 72 (d) 17
  30. 30.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 1156 (b) 1166 (c) 676 (d) 916
  31. 31.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9614 (b) 9996 (c) 10404 (d) 9604
  32. 32.If a + b = 13 and ab = 12, what is a² + b²? (a) 193 (b) 169 (c) 157 (d) 145
  33. 33.If a + b = 21 and ab = 108, what is a² + b²? (a) 441 (b) 657 (c) 333 (d) 225
  34. 34.Which is the correct factorisation of x² + 12x + 27? (a) (x + 3)(x + 9) (b) (x − 3)(x − 9) (c) (x + 1)(x + 11) (d) (x + 1)(x + 27)
  35. 35.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 9025 (b) 11035 (c) 10025 (d) 11025
  36. 36.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 794 (b) 1024 (c) 784 (d) 896
  37. 37.If a + b = 19 and ab = 90, what is a² + b²? (a) 271 (b) 181 (c) 361 (d) 541
  38. 38.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 2025 (b) 1225 (c) 1625 (d) 2035
  39. 39.If a + b = 10 and ab = 9, what is a² + b²? (a) 100 (b) 91 (c) 118 (d) 82
  40. 40.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 10) (b) (x − 2)(x − 5) (c) (x + 2)(x + 5) (d) (x + 1)(x + 6)

Answer key

  1. 160
  2. 6 and 7
  3. (x + 3)(x + 5)
  4. (x + 3)(x + 6)
  5. (x + 5)(x + 7)
  6. 100
  7. 5 and 8
  8. 1
  9. 1
  10. (x + 2)(x + 6)
  11. (x + 2)(x + 3)
  12. (x + 2)(x + 8)
  13. 13
  14. 36
  15. 7 and 8
  16. 16
  17. 12
  18. 6 and 9
  19. 7744
  20. 15
  21. 1
  22. 7056
  23. 1764
  24. 36
  25. 52
  26. 81
  27. 130
  28. 9
  29. 17
  30. 1156
  31. 9604
  32. 145
  33. 225
  34. (x + 3)(x + 9)
  35. 11025
  36. 784
  37. 181
  38. 2025
  39. 82
  40. (x + 2)(x + 5)

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