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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 235 · 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.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 12) (b) (x − 3)(x − 4) (c) (x + 3)(x + 4) (d) (x + 1)(x + 6)
  2. 2.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6734 (b) 6724 (c) 6404 (d) 6084
  3. 3.If a + b = 11 and ab = 24, what is a² + b²? (a) 121 (b) 97 (c) 169 (d) 73
  4. 4.What are the roots of x² − 11x + 30 = 0? (a) 5 and 6 (b) 2 and 15 (c) 1 and 30 (d) 1 and 10
  5. 5.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 6391 (b) 6889 (c) 5929 (d) 5939
  6. 6.Which is the correct factorisation of x² + 6x + 8? (a) (x − 2)(x − 4) (b) (x + 1)(x + 8) (c) (x + 1)(x + 5) (d) (x + 2)(x + 4)
  7. 7.What is the sum of the roots of x² − 13x + 42 = 0? (a) 12 (b) 13 (c) 14 (d) 42
  8. 8.What is the sum of the roots of x² − 13x + 40 = 0? (a) 14 (b) 40 (c) 13 (d) 12
  9. 9.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2819 (b) 2509 (c) 2209 (d) 2809
  10. 10.What is the sum of the roots of x² − 15x + 56 = 0? (a) 14 (b) 56 (c) 16 (d) 15
  11. 11.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 24) (b) (x + 3)(x + 8) (c) (x + 1)(x + 10) (d) (x − 3)(x − 8)
  12. 12.What is the sum of the roots of x² − 16x + 63 = 0? (a) 15 (b) 63 (c) 17 (d) 16
  13. 13.What are the roots of x² − 12x + 32 = 0? (a) 4 and 8 (b) 2 and 16 (c) 1 and 11 (d) 1 and 32
  14. 14.If a + b = 12 and ab = 27, what is (a − b)²? (a) 6 (b) 90 (c) 144 (d) 36
  15. 15.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3491 (b) 3721 (c) 3481 (d) 3599
  16. 16.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 8099 (b) 7921 (c) 8281 (d) 7931
  17. 17.What are the roots of x² − 15x + 56 = 0? (a) 2 and 28 (b) 1 and 14 (c) 1 and 56 (d) 7 and 8
  18. 18.If a + b = 16 and ab = 63, what is a² + b²? (a) 193 (b) 382 (c) 130 (d) 256
  19. 19.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10001 (b) 10201 (c) 9801 (d) 10211
  20. 20.What is the sum of the roots of x² − 15x + 54 = 0? (a) 16 (b) 14 (c) 54 (d) 15
  21. 21.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x − 8)(x − 9) (c) (x + 1)(x + 72) (d) (x + 8)(x + 9)
  22. 22.If a + b = 5 and ab = 6, what is a² + b²? (a) 37 (b) 13 (c) 25 (d) 19
  23. 23.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 416 (b) 576 (c) 256 (d) 586
  24. 24.What is the product of the roots of x² − 12x + 35 = 0? (a) 36 (b) 12 (c) 34 (d) 35
  25. 25.Which is the correct factorisation of x² + 11x + 30? (a) (x + 5)(x + 6) (b) (x + 1)(x + 10) (c) (x − 5)(x − 6) (d) (x + 1)(x + 30)
  26. 26.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 35) (b) (x − 5)(x − 7) (c) (x + 1)(x + 11) (d) (x + 5)(x + 7)
  27. 27.If a + b = 8 and ab = 7, what is a² + b²? (a) 78 (b) 57 (c) 64 (d) 50
  28. 28.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 72 (c) 17 (d) 73
  29. 29.What are the roots of x² − 8x + 15 = 0? (a) 1 and 7 (b) 2 and 6 (c) 1 and 15 (d) 3 and 5
  30. 30.If a + b = 17 and ab = 72, what is a² + b²? (a) 217 (b) 145 (c) 433 (d) 289
  31. 31.If a + b = 15 and ab = 56, what is a² + b²? (a) 337 (b) 113 (c) 169 (d) 225
  32. 32.What are the roots of x² − 16x + 63 = 0? (a) 7 and 9 (b) 3 and 21 (c) 1 and 15 (d) 1 and 63
  33. 33.If a + b = 13 and ab = 22, what is (a − b)²? (a) 9 (b) 125 (c) 81 (d) 169
  34. 34.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 4096 (b) 3616 (c) 4106 (d) 3136
  35. 35.What is the sum of the roots of x² − 11x + 24 = 0? (a) 11 (b) 24 (c) 10 (d) 12
  36. 36.If a + b = 14 and ab = 48, what is a² + b²? (a) 196 (b) 148 (c) 292 (d) 100
  37. 37.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 375 (b) 235 (c) 625 (d) 225
  38. 38.What are the roots of x² − 17x + 72 = 0? (a) 2 and 36 (b) 8 and 9 (c) 1 and 72 (d) 1 and 16
  39. 39.What are the roots of x² − 8x + 12 = 0? (a) 1 and 7 (b) 3 and 4 (c) 2 and 6 (d) 1 and 12
  40. 40.If a + b = 19 and ab = 88, what is (a − b)²? (a) 3 (b) 9 (c) 361 (d) 185

Answer key

  1. (x + 3)(x + 4)
  2. 6724
  3. 73
  4. 5 and 6
  5. 5929
  6. (x + 2)(x + 4)
  7. 13
  8. 13
  9. 2809
  10. 15
  11. (x + 3)(x + 8)
  12. 16
  13. 4 and 8
  14. 36
  15. 3481
  16. 7921
  17. 7 and 8
  18. 130
  19. 10201
  20. 15
  21. (x + 8)(x + 9)
  22. 13
  23. 576
  24. 35
  25. (x + 5)(x + 6)
  26. (x + 5)(x + 7)
  27. 50
  28. 72
  29. 3 and 5
  30. 145
  31. 113
  32. 7 and 9
  33. 81
  34. 4096
  35. 11
  36. 100
  37. 225
  38. 8 and 9
  39. 2 and 6
  40. 9

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