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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 245 · 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.What is the product of the roots of x² − 11x + 30 = 0? (a) 30 (b) 29 (c) 31 (d) 11
  2. 2.If a + b = 8 and ab = 15, what is a² + b²? (a) 94 (b) 49 (c) 64 (d) 34
  3. 3.If a + b = 9 and ab = 20, what is a² + b²? (a) 81 (b) 61 (c) 121 (d) 41
  4. 4.What are the roots of x² − 15x + 56 = 0? (a) 7 and 8 (b) 1 and 56 (c) 2 and 28 (d) 1 and 14
  5. 5.If a + b = 13 and ab = 42, what is a² + b²? (a) 85 (b) 253 (c) 169 (d) 127
  6. 6.If a + b = 20 and ab = 99, what is (a − b)²? (a) 2 (b) 202 (c) 4 (d) 400
  7. 7.If a + b = 3 and ab = 2, what is (a − b)²? (a) 7 (b) 5 (c) 1 (d) 9
  8. 8.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 5786 (b) 7056 (c) 6384 (d) 5776
  9. 9.Which is the correct factorisation of x² + 8x + 12? (a) (x + 2)(x + 6) (b) (x + 1)(x + 12) (c) (x + 1)(x + 7) (d) (x − 2)(x − 6)
  10. 10.What is the sum of the roots of x² − 15x + 54 = 0? (a) 14 (b) 15 (c) 16 (d) 54
  11. 11.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3364 (b) 3844 (c) 3596 (d) 3374
  12. 12.Which is the correct factorisation of x² + 11x + 28? (a) (x − 4)(x − 7) (b) (x + 1)(x + 10) (c) (x + 1)(x + 28) (d) (x + 4)(x + 7)
  13. 13.If a + b = 18 and ab = 77, what is a² + b²? (a) 478 (b) 247 (c) 324 (d) 170
  14. 14.If a + b = 17 and ab = 60, what is a² + b²? (a) 289 (b) 409 (c) 169 (d) 229
  15. 15.If a + b = 13 and ab = 40, what is a² + b²? (a) 129 (b) 89 (c) 169 (d) 249
  16. 16.What is the product of the roots of x² − 8x + 12 = 0? (a) 12 (b) 11 (c) 13 (d) 8
  17. 17.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 35) (b) (x + 5)(x + 7) (c) (x − 5)(x − 7) (d) (x + 1)(x + 11)
  18. 18.What are the roots of x² − 13x + 40 = 0? (a) 2 and 20 (b) 1 and 40 (c) 1 and 12 (d) 5 and 8
  19. 19.If a + b = 12 and ab = 20, what is a² + b²? (a) 104 (b) 124 (c) 184 (d) 144
  20. 20.What are the roots of x² − 10x + 16 = 0? (a) 1 and 16 (b) 3 and 7 (c) 1 and 9 (d) 2 and 8
  21. 21.If a + b = 15 and ab = 56, what is (a − b)²? (a) 169 (b) 225 (c) 1 (d) 113
  22. 22.If a + b = 18 and ab = 72, what is (a − b)²? (a) 324 (b) 36 (c) 6 (d) 180
  23. 23.If a + b = 11 and ab = 30, what is a² + b²? (a) 91 (b) 181 (c) 61 (d) 121
  24. 24.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8659 (b) 7569 (c) 8109 (d) 8649
  25. 25.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 676 (b) 884 (c) 1156 (d) 686
  26. 26.What are the roots of x² − 17x + 72 = 0? (a) 8 and 9 (b) 1 and 72 (c) 1 and 16 (d) 2 and 36
  27. 27.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9996 (b) 10404 (c) 9604 (d) 9614
  28. 28.What is the product of the roots of x² − 14x + 45 = 0? (a) 46 (b) 44 (c) 14 (d) 45
  29. 29.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 3616 (b) 4106 (c) 3136 (d) 4096
  30. 30.If a + b = 19 and ab = 88, what is a² + b²? (a) 185 (b) 361 (c) 273 (d) 537
  31. 31.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x + 1)(x + 56) (c) (x − 7)(x − 8) (d) (x + 7)(x + 8)
  32. 32.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 784 (b) 794 (c) 1024 (d) 896
  33. 33.What are the roots of x² − 11x + 18 = 0? (a) 3 and 6 (b) 2 and 9 (c) 1 and 10 (d) 1 and 18
  34. 34.Which is the correct factorisation of x² + 11x + 30? (a) (x + 5)(x + 6) (b) (x + 1)(x + 30) (c) (x − 5)(x − 6) (d) (x + 1)(x + 10)
  35. 35.If a + b = 14 and ab = 40, what is (a − b)²? (a) 36 (b) 196 (c) 6 (d) 116
  36. 36.If a + b = 12 and ab = 11, what is (a − b)²? (a) 100 (b) 10 (c) 144 (d) 122
  37. 37.Which is the correct factorisation of x² + 10x + 16? (a) (x + 2)(x + 8) (b) (x − 2)(x − 8) (c) (x + 1)(x + 16) (d) (x + 1)(x + 9)
  38. 38.If a + b = 7 and ab = 12, what is a² + b²? (a) 37 (b) 25 (c) 73 (d) 49
  39. 39.If a + b = 7 and ab = 10, what is (a − b)²? (a) 3 (b) 9 (c) 29 (d) 49
  40. 40.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x + 1)(x + 42) (c) (x − 6)(x − 7) (d) (x + 6)(x + 7)

Answer key

  1. 30
  2. 34
  3. 41
  4. 7 and 8
  5. 85
  6. 4
  7. 1
  8. 5776
  9. (x + 2)(x + 6)
  10. 15
  11. 3364
  12. (x + 4)(x + 7)
  13. 170
  14. 169
  15. 89
  16. 12
  17. (x + 5)(x + 7)
  18. 5 and 8
  19. 104
  20. 2 and 8
  21. 1
  22. 36
  23. 61
  24. 8649
  25. 676
  26. 8 and 9
  27. 9604
  28. 45
  29. 4096
  30. 185
  31. (x + 7)(x + 8)
  32. 784
  33. 2 and 9
  34. (x + 5)(x + 6)
  35. 36
  36. 100
  37. (x + 2)(x + 8)
  38. 25
  39. 9
  40. (x + 6)(x + 7)

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