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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 875 · 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 + 13x + 36? (a) (x + 1)(x + 12) (b) (x + 1)(x + 36) (c) (x + 4)(x + 9) (d) (x − 4)(x − 9)
  2. 2.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)
  3. 3.If a + b = 13 and ab = 12, what is a² + b²? (a) 169 (b) 157 (c) 193 (d) 145
  4. 4.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 8) (b) (x + 3)(x + 6) (c) (x − 3)(x − 6) (d) (x + 1)(x + 18)
  5. 5.If a + b = 15 and ab = 50, what is (a − b)²? (a) 5 (b) 25 (c) 225 (d) 125
  6. 6.If a + b = 12 and ab = 32, what is (a − b)²? (a) 16 (b) 144 (c) 80 (d) 4
  7. 7.If a + b = 18 and ab = 77, what is (a − b)²? (a) 16 (b) 170 (c) 324 (d) 4
  8. 8.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 8) (b) (x + 1)(x + 5) (c) (x − 2)(x − 4) (d) (x + 2)(x + 4)
  9. 9.If a + b = 17 and ab = 66, what is a² + b²? (a) 289 (b) 223 (c) 157 (d) 421
  10. 10.If a + b = 15 and ab = 54, what is a² + b²? (a) 117 (b) 225 (c) 171 (d) 333
  11. 11.What are the roots of x² − 14x + 45 = 0? (a) 1 and 45 (b) 3 and 15 (c) 1 and 13 (d) 5 and 9
  12. 12.If a + b = 18 and ab = 72, what is a² + b²? (a) 180 (b) 468 (c) 324 (d) 252
  13. 13.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 7066 (b) 7056 (c) 5776 (d) 6416
  14. 14.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 8 and 9 (c) 1 and 16 (d) 2 and 36
  15. 15.What is the sum of the roots of x² − 5x + 6 = 0? (a) 6 (b) 4 (c) 7 (d) 5
  16. 16.If a + b = 14 and ab = 45, what is (a − b)²? (a) 106 (b) 196 (c) 16 (d) 4
  17. 17.Which is the correct factorisation of x² + 5x + 6? (a) (x − 2)(x − 3) (b) (x + 2)(x + 3) (c) (x + 1)(x + 6) (d) (x + 1)(x + 4)
  18. 18.If a + b = 10 and ab = 21, what is (a − b)²? (a) 4 (b) 100 (c) 58 (d) 16
  19. 19.If a + b = 16 and ab = 55, what is a² + b²? (a) 146 (b) 201 (c) 366 (d) 256
  20. 20.If a + b = 13 and ab = 36, what is a² + b²? (a) 133 (b) 241 (c) 97 (d) 169
  21. 21.What is the sum of the roots of x² − 12x + 35 = 0? (a) 35 (b) 13 (c) 12 (d) 11
  22. 22.Which is the correct factorisation of x² + 10x + 24? (a) (x − 4)(x − 6) (b) (x + 4)(x + 6) (c) (x + 1)(x + 24) (d) (x + 1)(x + 9)
  23. 23.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 12) (b) (x − 2)(x − 6) (c) (x + 1)(x + 7) (d) (x + 2)(x + 6)
  24. 24.If a + b = 9 and ab = 18, what is (a − b)²? (a) 45 (b) 81 (c) 3 (d) 9
  25. 25.Which is the correct factorisation of x² + 10x + 16? (a) (x − 2)(x − 8) (b) (x + 1)(x + 9) (c) (x + 1)(x + 16) (d) (x + 2)(x + 8)
  26. 26.If a + b = 15 and ab = 56, what is (a − b)²? (a) 169 (b) 113 (c) 225 (d) 1
  27. 27.If a + b = 10 and ab = 9, what is a² + b²? (a) 100 (b) 82 (c) 118 (d) 91
  28. 28.If a + b = 18 and ab = 72, what is (a − b)²? (a) 324 (b) 180 (c) 6 (d) 36
  29. 29.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1521 (b) 1601 (c) 1691 (d) 1681
  30. 30.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 18 (c) 17 (d) 16
  31. 31.If a + b = 19 and ab = 84, what is a² + b²? (a) 277 (b) 361 (c) 529 (d) 193
  32. 32.If a + b = 11 and ab = 24, what is (a − b)²? (a) 25 (b) 5 (c) 121 (d) 73
  33. 33.If a + b = 9 and ab = 14, what is a² + b²? (a) 67 (b) 81 (c) 53 (d) 109
  34. 34.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 9984 (b) 10816 (c) 9226 (d) 9216
  35. 35.If a + b = 20 and ab = 99, what is (a − b)²? (a) 4 (b) 2 (c) 202 (d) 400
  36. 36.What are the roots of x² − 10x + 21 = 0? (a) 1 and 9 (b) 2 and 8 (c) 1 and 21 (d) 3 and 7
  37. 37.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2601 (b) 2401 (c) 2411 (d) 2499
  38. 38.If a + b = 19 and ab = 84, what is (a − b)²? (a) 25 (b) 5 (c) 361 (d) 193
  39. 39.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 28) (b) (x − 4)(x − 7) (c) (x + 4)(x + 7) (d) (x + 1)(x + 10)
  40. 40.Which is the correct factorisation of x² + 11x + 24? (a) (x − 3)(x − 8) (b) (x + 1)(x + 24) (c) (x + 3)(x + 8) (d) (x + 1)(x + 10)

Answer key

  1. (x + 4)(x + 9)
  2. (x + 8)(x + 9)
  3. 145
  4. (x + 3)(x + 6)
  5. 25
  6. 16
  7. 16
  8. (x + 2)(x + 4)
  9. 157
  10. 117
  11. 5 and 9
  12. 180
  13. 7056
  14. 8 and 9
  15. 5
  16. 16
  17. (x + 2)(x + 3)
  18. 16
  19. 146
  20. 97
  21. 12
  22. (x + 4)(x + 6)
  23. (x + 2)(x + 6)
  24. 9
  25. (x + 2)(x + 8)
  26. 1
  27. 82
  28. 36
  29. 1681
  30. 17
  31. 193
  32. 25
  33. 53
  34. 9216
  35. 4
  36. 3 and 7
  37. 2401
  38. 25
  39. (x + 4)(x + 7)
  40. (x + 3)(x + 8)

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