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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 887 · 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 = 12, what is a² + b²? (a) 193 (b) 145 (c) 169 (d) 157
  2. 2.If a + b = 20 and ab = 99, what is (a − b)²? (a) 2 (b) 4 (c) 202 (d) 400
  3. 3.What is the sum of the roots of x² − 8x + 15 = 0? (a) 8 (b) 7 (c) 15 (d) 9
  4. 4.If a + b = 3 and ab = 2, what is (a − b)²? (a) 7 (b) 5 (c) 1 (d) 9
  5. 5.If a + b = 21 and ab = 108, what is (a − b)²? (a) 441 (b) 225 (c) 3 (d) 9
  6. 6.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 14) (b) (x + 1)(x + 8) (c) (x − 2)(x − 7) (d) (x + 2)(x + 7)
  7. 7.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x + 1)(x + 13) (c) (x + 6)(x + 8) (d) (x − 6)(x − 8)
  8. 8.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 225 (b) 625 (c) 635 (d) 425
  9. 9.If a + b = 4 and ab = 3, what is a² + b²? (a) 10 (b) 13 (c) 16 (d) 22
  10. 10.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 16) (c) (x + 8)(x + 9) (d) (x + 1)(x + 72)
  11. 11.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1609 (b) 1849 (c) 1859 (d) 1369
  12. 12.If a + b = 15 and ab = 36, what is a² + b²? (a) 153 (b) 225 (c) 297 (d) 189
  13. 13.If a + b = 10 and ab = 9, what is (a − b)²? (a) 82 (b) 100 (c) 8 (d) 64
  14. 14.Which is the correct factorisation of x² + 10x + 21? (a) (x − 3)(x − 7) (b) (x + 1)(x + 9) (c) (x + 1)(x + 21) (d) (x + 3)(x + 7)
  15. 15.If a + b = 14 and ab = 48, what is (a − b)²? (a) 100 (b) 196 (c) 4 (d) 2
  16. 16.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 5184 (b) 4896 (c) 4634 (d) 4624
  17. 17.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 28) (b) (x + 1)(x + 10) (c) (x − 4)(x − 7) (d) (x + 4)(x + 7)
  18. 18.Which is the correct factorisation of x² + 8x + 15? (a) (x − 3)(x − 5) (b) (x + 3)(x + 5) (c) (x + 1)(x + 15) (d) (x + 1)(x + 7)
  19. 19.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8101 (b) 8281 (c) 8291 (d) 7921
  20. 20.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 7754 (b) 7744 (c) 8464 (d) 8096
  21. 21.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6724 (b) 6084 (c) 6734 (d) 6404
  22. 22.If a + b = 11 and ab = 24, what is (a − b)²? (a) 25 (b) 5 (c) 73 (d) 121
  23. 23.What is the product of the roots of x² − 8x + 15 = 0? (a) 16 (b) 8 (c) 15 (d) 14
  24. 24.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 399 (b) 361 (c) 371 (d) 441
  25. 25.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 7)(x + 9) (c) (x − 7)(x − 9) (d) (x + 1)(x + 63)
  26. 26.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1599 (b) 1521 (c) 1681 (d) 1531
  27. 27.If a + b = 7 and ab = 10, what is a² + b²? (a) 49 (b) 69 (c) 29 (d) 39
  28. 28.If a + b = 3 and ab = 2, what is a² + b²? (a) 5 (b) 7 (c) 13 (d) 9
  29. 29.If a + b = 12 and ab = 35, what is a² + b²? (a) 214 (b) 109 (c) 74 (d) 144
  30. 30.If a + b = 14 and ab = 48, what is a² + b²? (a) 196 (b) 148 (c) 100 (d) 292
  31. 31.What are the roots of x² − 11x + 18 = 0? (a) 3 and 6 (b) 1 and 18 (c) 1 and 10 (d) 2 and 9
  32. 32.If a + b = 6 and ab = 5, what is (a − b)²? (a) 16 (b) 4 (c) 36 (d) 26
  33. 33.What are the roots of x² − 15x + 54 = 0? (a) 1 and 14 (b) 2 and 27 (c) 1 and 54 (d) 6 and 9
  34. 34.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 299 (b) 391 (c) 529 (d) 289
  35. 35.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 794 (b) 784 (c) 896 (d) 1024
  36. 36.What is the product of the roots of x² − 14x + 45 = 0? (a) 46 (b) 14 (c) 45 (d) 44
  37. 37.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 2 and 36 (c) 1 and 72 (d) 8 and 9
  38. 38.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 8836 (b) 7396 (c) 8084 (d) 7406
  39. 39.What is the product of the roots of x² − 16x + 63 = 0? (a) 64 (b) 16 (c) 63 (d) 62
  40. 40.If a + b = 12 and ab = 32, what is (a − b)²? (a) 144 (b) 16 (c) 80 (d) 4

Answer key

  1. 145
  2. 4
  3. 8
  4. 1
  5. 9
  6. (x + 2)(x + 7)
  7. (x + 6)(x + 8)
  8. 625
  9. 10
  10. (x + 8)(x + 9)
  11. 1849
  12. 153
  13. 64
  14. (x + 3)(x + 7)
  15. 4
  16. 4624
  17. (x + 4)(x + 7)
  18. (x + 3)(x + 5)
  19. 8281
  20. 7744
  21. 6724
  22. 25
  23. 15
  24. 361
  25. (x + 7)(x + 9)
  26. 1521
  27. 29
  28. 5
  29. 74
  30. 100
  31. 2 and 9
  32. 16
  33. 6 and 9
  34. 289
  35. 784
  36. 45
  37. 8 and 9
  38. 7396
  39. 63
  40. 16

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