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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 328 · 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 68²? (a) 4624 (b) 4896 (c) 4634 (d) 5184
  2. 2.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x − 7)(x − 9) (c) (x + 1)(x + 15) (d) (x + 7)(x + 9)
  3. 3.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 11) (b) (x − 3)(x − 9) (c) (x + 1)(x + 27) (d) (x + 3)(x + 9)
  4. 4.If a + b = 15 and ab = 36, what is (a − b)²? (a) 81 (b) 153 (c) 9 (d) 225
  5. 5.If a + b = 8 and ab = 15, what is (a − b)²? (a) 4 (b) 64 (c) 2 (d) 34
  6. 6.What is the sum of the roots of x² − 10x + 24 = 0? (a) 24 (b) 10 (c) 11 (d) 9
  7. 7.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1369 (b) 1859 (c) 1609 (d) 1849
  8. 8.If a + b = 19 and ab = 84, what is (a − b)²? (a) 5 (b) 193 (c) 25 (d) 361
  9. 9.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 18 (c) 17 (d) 16
  10. 10.If a + b = 19 and ab = 84, what is a² + b²? (a) 277 (b) 529 (c) 361 (d) 193
  11. 11.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 10) (b) (x + 1)(x + 6) (c) (x − 2)(x − 5) (d) (x + 2)(x + 5)
  12. 12.If a + b = 6 and ab = 5, what is (a − b)²? (a) 26 (b) 16 (c) 36 (d) 4
  13. 13.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)
  14. 14.If a + b = 20 and ab = 96, what is (a − b)²? (a) 208 (b) 4 (c) 400 (d) 16
  15. 15.If a + b = 15 and ab = 54, what is a² + b²? (a) 117 (b) 171 (c) 333 (d) 225
  16. 16.What is the product of the roots of x² − 5x + 6 = 0? (a) 7 (b) 8 (c) 5 (d) 6
  17. 17.If a + b = 14 and ab = 45, what is a² + b²? (a) 151 (b) 106 (c) 286 (d) 196
  18. 18.What is the product of the roots of x² − 14x + 48 = 0? (a) 49 (b) 48 (c) 14 (d) 47
  19. 19.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1454 (b) 1596 (c) 1444 (d) 1764
  20. 20.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 3025 (b) 3035 (c) 3575 (d) 4225
  21. 21.If a + b = 13 and ab = 42, what is (a − b)²? (a) 169 (b) 1 (c) 127 (d) 85
  22. 22.What is the product of the roots of x² − 7x + 12 = 0? (a) 11 (b) 12 (c) 13 (d) 7
  23. 23.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 539 (b) 289 (c) 529 (d) 409
  24. 24.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3731 (b) 3481 (c) 3601 (d) 3721
  25. 25.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 1)(x + 13) (c) (x + 6)(x + 8) (d) (x + 1)(x + 48)
  26. 26.What is the product of the roots of x² − 8x + 12 = 0? (a) 13 (b) 11 (c) 12 (d) 8
  27. 27.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 9216 (b) 9226 (c) 9984 (d) 10816
  28. 28.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 8836 (b) 8846 (c) 8116 (d) 7396
  29. 29.Which is the correct factorisation of x² + 7x + 12? (a) (x + 3)(x + 4) (b) (x + 1)(x + 12) (c) (x − 3)(x − 4) (d) (x + 1)(x + 6)
  30. 30.If a + b = 10 and ab = 9, what is a² + b²? (a) 82 (b) 100 (c) 91 (d) 118
  31. 31.If a + b = 16 and ab = 60, what is a² + b²? (a) 196 (b) 136 (c) 376 (d) 256
  32. 32.If a + b = 10 and ab = 16, what is (a − b)²? (a) 6 (b) 68 (c) 100 (d) 36
  33. 33.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 5476 (b) 4356 (c) 4916 (d) 5486
  34. 34.If a + b = 9 and ab = 18, what is a² + b²? (a) 117 (b) 63 (c) 45 (d) 81
  35. 35.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 12) (b) (x − 4)(x − 9) (c) (x + 4)(x + 9) (d) (x + 1)(x + 36)
  36. 36.What is the product of the roots of x² − 11x + 24 = 0? (a) 24 (b) 11 (c) 25 (d) 23
  37. 37.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 9) (b) (x + 3)(x + 7) (c) (x − 3)(x − 7) (d) (x + 1)(x + 21)
  38. 38.If a + b = 15 and ab = 50, what is a² + b²? (a) 225 (b) 175 (c) 125 (d) 325
  39. 39.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 256 (b) 266 (c) 384 (d) 576
  40. 40.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3969 (b) 3609 (c) 3979 (d) 3249

Answer key

  1. 4624
  2. (x + 7)(x + 9)
  3. (x + 3)(x + 9)
  4. 81
  5. 4
  6. 10
  7. 1849
  8. 25
  9. 17
  10. 193
  11. (x + 2)(x + 5)
  12. 16
  13. (x + 8)(x + 9)
  14. 16
  15. 117
  16. 6
  17. 106
  18. 48
  19. 1444
  20. 3025
  21. 1
  22. 12
  23. 529
  24. 3721
  25. (x + 6)(x + 8)
  26. 12
  27. 9216
  28. 8836
  29. (x + 3)(x + 4)
  30. 82
  31. 136
  32. 36
  33. 5476
  34. 45
  35. (x + 4)(x + 9)
  36. 24
  37. (x + 3)(x + 7)
  38. 125
  39. 256
  40. 3969

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