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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 718 · 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.If a + b = 13 and ab = 30, what is (a − b)²? (a) 49 (b) 169 (c) 109 (d) 7
  2. 2.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 8464 (b) 7744 (c) 8096 (d) 7754
  3. 3.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 8091 (b) 7579 (c) 8649 (d) 7569
  4. 4.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3599 (b) 3481 (c) 3721 (d) 3491
  5. 5.What are the roots of x² − 15x + 56 = 0? (a) 2 and 28 (b) 1 and 56 (c) 7 and 8 (d) 1 and 14
  6. 6.If a + b = 6 and ab = 8, what is a² + b²? (a) 36 (b) 52 (c) 28 (d) 20
  7. 7.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x − 8)(x − 9) (c) (x + 8)(x + 9) (d) (x + 1)(x + 16)
  8. 8.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3731 (b) 3481 (c) 3601 (d) 3721
  9. 9.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 11035 (b) 10025 (c) 11025 (d) 9025
  10. 10.What is the sum of the roots of x² − 9x + 14 = 0? (a) 10 (b) 8 (c) 14 (d) 9
  11. 11.What are the roots of x² − 10x + 16 = 0? (a) 1 and 16 (b) 1 and 9 (c) 2 and 8 (d) 3 and 7
  12. 12.What is the product of the roots of x² − 7x + 12 = 0? (a) 11 (b) 7 (c) 13 (d) 12
  13. 13.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 7)(x + 8) (c) (x − 7)(x − 8) (d) (x + 1)(x + 14)
  14. 14.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 5476 (b) 4356 (c) 4916 (d) 5486
  15. 15.Which is the correct factorisation of x² + 8x + 15? (a) (x − 3)(x − 5) (b) (x + 1)(x + 7) (c) (x + 1)(x + 15) (d) (x + 3)(x + 5)
  16. 16.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)
  17. 17.What are the roots of x² − 14x + 48 = 0? (a) 1 and 48 (b) 2 and 24 (c) 6 and 8 (d) 1 and 13
  18. 18.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 45) (b) (x − 5)(x − 9) (c) (x + 5)(x + 9) (d) (x + 1)(x + 13)
  19. 19.Which is the correct factorisation of x² + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 3)(x + 4) (c) (x + 1)(x + 12) (d) (x + 1)(x + 6)
  20. 20.What is the sum of the roots of x² − 14x + 48 = 0? (a) 15 (b) 14 (c) 48 (d) 13
  21. 21.If a + b = 12 and ab = 35, what is (a − b)²? (a) 74 (b) 144 (c) 2 (d) 4
  22. 22.If a + b = 14 and ab = 48, what is (a − b)²? (a) 2 (b) 100 (c) 196 (d) 4
  23. 23.If a + b = 17 and ab = 66, what is a² + b²? (a) 223 (b) 157 (c) 289 (d) 421
  24. 24.If a + b = 12 and ab = 35, what is a² + b²? (a) 109 (b) 74 (c) 144 (d) 214
  25. 25.If a + b = 16 and ab = 63, what is (a − b)²? (a) 2 (b) 256 (c) 4 (d) 130
  26. 26.If a + b = 15 and ab = 36, what is a² + b²? (a) 153 (b) 189 (c) 297 (d) 225
  27. 27.If a + b = 10 and ab = 21, what is a² + b²? (a) 79 (b) 142 (c) 100 (d) 58
  28. 28.What are the roots of x² − 12x + 35 = 0? (a) 2 and 10 (b) 5 and 7 (c) 1 and 11 (d) 1 and 35
  29. 29.If a + b = 21 and ab = 108, what is (a − b)²? (a) 441 (b) 3 (c) 225 (d) 9
  30. 30.What is the product of the roots of x² − 12x + 35 = 0? (a) 34 (b) 12 (c) 35 (d) 36
  31. 31.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 10201 (b) 9811 (c) 9801 (d) 9999
  32. 32.If a + b = 10 and ab = 21, what is (a − b)²? (a) 100 (b) 4 (c) 58 (d) 16
  33. 33.What is the sum of the roots of x² − 15x + 54 = 0? (a) 14 (b) 16 (c) 15 (d) 54
  34. 34.If a + b = 21 and ab = 108, what is a² + b²? (a) 225 (b) 441 (c) 333 (d) 657
  35. 35.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3596 (b) 3374 (c) 3364 (d) 3844
  36. 36.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9025 (b) 9975 (c) 11025 (d) 9035
  37. 37.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1089 (b) 909 (c) 729 (d) 1099
  38. 38.If a + b = 19 and ab = 88, what is (a − b)²? (a) 185 (b) 3 (c) 361 (d) 9
  39. 39.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 4925 (b) 4225 (c) 5625 (d) 5635
  40. 40.If a + b = 4 and ab = 3, what is a² + b²? (a) 13 (b) 10 (c) 22 (d) 16

Answer key

  1. 49
  2. 7744
  3. 7569
  4. 3481
  5. 7 and 8
  6. 20
  7. (x + 8)(x + 9)
  8. 3721
  9. 11025
  10. 9
  11. 2 and 8
  12. 12
  13. (x + 7)(x + 8)
  14. 5476
  15. (x + 3)(x + 5)
  16. (x + 6)(x + 9)
  17. 6 and 8
  18. (x + 5)(x + 9)
  19. (x + 3)(x + 4)
  20. 14
  21. 4
  22. 4
  23. 157
  24. 74
  25. 4
  26. 153
  27. 58
  28. 5 and 7
  29. 9
  30. 35
  31. 9801
  32. 16
  33. 15
  34. 225
  35. 3364
  36. 9025
  37. 1089
  38. 9
  39. 5625
  40. 10

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