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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 691 · 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 = 8 and ab = 15, what is a² + b²? (a) 49 (b) 94 (c) 64 (d) 34
  2. 2.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1764 (b) 1604 (c) 1774 (d) 1444
  3. 3.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 1)(x + 63) (c) (x − 7)(x − 9) (d) (x + 7)(x + 9)
  4. 4.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 32) (b) (x + 1)(x + 11) (c) (x − 4)(x − 8) (d) (x + 4)(x + 8)
  5. 5.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 3035 (b) 4225 (c) 3025 (d) 3575
  6. 6.If a + b = 4 and ab = 3, what is a² + b²? (a) 13 (b) 22 (c) 10 (d) 16
  7. 7.What is the product of the roots of x² − 16x + 63 = 0? (a) 63 (b) 64 (c) 62 (d) 16
  8. 8.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8281 (b) 7921 (c) 8291 (d) 8101
  9. 9.What are the roots of x² − 13x + 36 = 0? (a) 4 and 9 (b) 1 and 36 (c) 2 and 18 (d) 1 and 12
  10. 10.If a + b = 7 and ab = 12, what is a² + b²? (a) 37 (b) 25 (c) 73 (d) 49
  11. 11.If a + b = 4 and ab = 3, what is (a − b)²? (a) 4 (b) 10 (c) 16 (d) 2
  12. 12.What are the roots of x² − 6x + 8 = 0? (a) 2 and 5 (b) 1 and 5 (c) 2 and 4 (d) 1 and 8
  13. 13.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)
  14. 14.What is the product of the roots of x² − 13x + 36 = 0? (a) 36 (b) 13 (c) 37 (d) 35
  15. 15.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 529 (b) 391 (c) 289 (d) 299
  16. 16.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x − 6)(x − 8) (c) (x + 1)(x + 13) (d) (x + 6)(x + 8)
  17. 17.If a + b = 18 and ab = 72, what is a² + b²? (a) 324 (b) 468 (c) 180 (d) 252
  18. 18.If a + b = 6 and ab = 8, what is (a − b)²? (a) 4 (b) 2 (c) 20 (d) 36
  19. 19.What is the sum of the roots of x² − 15x + 54 = 0? (a) 14 (b) 16 (c) 15 (d) 54
  20. 20.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 841 (b) 901 (c) 971 (d) 961
  21. 21.Which is the correct factorisation of x² + 5x + 6? (a) (x + 2)(x + 3) (b) (x + 1)(x + 6) (c) (x + 1)(x + 4) (d) (x − 2)(x − 3)
  22. 22.What are the roots of x² − 15x + 56 = 0? (a) 1 and 14 (b) 2 and 28 (c) 7 and 8 (d) 1 and 56
  23. 23.If a + b = 21 and ab = 108, what is (a − b)²? (a) 225 (b) 441 (c) 9 (d) 3
  24. 24.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x − 6)(x − 7) (c) (x + 1)(x + 42) (d) (x + 6)(x + 7)
  25. 25.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 529 (b) 289 (c) 539 (d) 409
  26. 26.If a + b = 13 and ab = 12, what is a² + b²? (a) 169 (b) 145 (c) 157 (d) 193
  27. 27.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 73 (c) 72 (d) 71
  28. 28.What are the roots of x² − 10x + 24 = 0? (a) 1 and 9 (b) 2 and 12 (c) 4 and 6 (d) 1 and 24
  29. 29.What is the product of the roots of x² − 11x + 30 = 0? (a) 31 (b) 29 (c) 11 (d) 30
  30. 30.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 8)(x + 9) (c) (x + 1)(x + 16) (d) (x − 8)(x − 9)
  31. 31.What is the sum of the roots of x² − 10x + 24 = 0? (a) 9 (b) 10 (c) 24 (d) 11
  32. 32.If a + b = 13 and ab = 30, what is a² + b²? (a) 169 (b) 139 (c) 109 (d) 229
  33. 33.What is the sum of the roots of x² − 7x + 12 = 0? (a) 7 (b) 6 (c) 12 (d) 8
  34. 34.What are the roots of x² − 13x + 42 = 0? (a) 1 and 42 (b) 2 and 21 (c) 1 and 12 (d) 6 and 7
  35. 35.What is the sum of the roots of x² − 5x + 6 = 0? (a) 6 (b) 5 (c) 4 (d) 7
  36. 36.If a + b = 12 and ab = 27, what is (a − b)²? (a) 90 (b) 144 (c) 6 (d) 36
  37. 37.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1599 (b) 1681 (c) 1521 (d) 1531
  38. 38.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 7744 (b) 8104 (c) 8474 (d) 8464
  39. 39.What are the roots of x² − 14x + 45 = 0? (a) 1 and 13 (b) 5 and 9 (c) 1 and 45 (d) 3 and 15
  40. 40.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1764 (b) 1444 (c) 1454 (d) 1596

Answer key

  1. 34
  2. 1764
  3. (x + 7)(x + 9)
  4. (x + 4)(x + 8)
  5. 3025
  6. 10
  7. 63
  8. 8281
  9. 4 and 9
  10. 25
  11. 4
  12. 2 and 4
  13. (x + 3)(x + 9)
  14. 36
  15. 289
  16. (x + 6)(x + 8)
  17. 180
  18. 4
  19. 15
  20. 961
  21. (x + 2)(x + 3)
  22. 7 and 8
  23. 9
  24. (x + 6)(x + 7)
  25. 529
  26. 145
  27. 72
  28. 4 and 6
  29. 30
  30. (x + 8)(x + 9)
  31. 10
  32. 109
  33. 7
  34. 6 and 7
  35. 5
  36. 36
  37. 1521
  38. 8464
  39. 5 and 9
  40. 1444

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