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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 473 · 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 + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 7)(x + 8) (c) (x + 1)(x + 14) (d) (x + 1)(x + 56)
  2. 2.If a + b = 16 and ab = 63, what is a² + b²? (a) 256 (b) 382 (c) 193 (d) 130
  3. 3.If a + b = 18 and ab = 72, what is a² + b²? (a) 324 (b) 180 (c) 468 (d) 252
  4. 4.If a + b = 14 and ab = 24, what is a² + b²? (a) 148 (b) 196 (c) 172 (d) 244
  5. 5.What is the sum of the roots of x² − 16x + 63 = 0? (a) 16 (b) 63 (c) 17 (d) 15
  6. 6.If a + b = 13 and ab = 12, what is a² + b²? (a) 169 (b) 157 (c) 145 (d) 193
  7. 7.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 1166 (b) 916 (c) 676 (d) 1156
  8. 8.If a + b = 5 and ab = 4, what is a² + b²? (a) 17 (b) 25 (c) 21 (d) 33
  9. 9.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 6) (b) (x + 1)(x + 4) (c) (x − 2)(x − 3) (d) (x + 2)(x + 3)
  10. 10.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x + 1)(x + 72) (c) (x + 1)(x + 16) (d) (x − 8)(x − 9)
  11. 11.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 3146 (b) 3584 (c) 4096 (d) 3136
  12. 12.If a + b = 9 and ab = 20, what is (a − b)²? (a) 41 (b) 61 (c) 81 (d) 1
  13. 13.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1774 (b) 1764 (c) 1604 (d) 1444
  14. 14.If a + b = 19 and ab = 90, what is (a − b)²? (a) 1 (b) 271 (c) 181 (d) 361
  15. 15.If a + b = 13 and ab = 12, what is (a − b)²? (a) 11 (b) 169 (c) 145 (d) 121
  16. 16.What is the product of the roots of x² − 7x + 12 = 0? (a) 7 (b) 11 (c) 12 (d) 13
  17. 17.If a + b = 14 and ab = 48, what is (a − b)²? (a) 4 (b) 196 (c) 100 (d) 2
  18. 18.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 8649 (b) 7579 (c) 8091 (d) 7569
  19. 19.If a + b = 11 and ab = 28, what is a² + b²? (a) 177 (b) 121 (c) 93 (d) 65
  20. 20.Which is the correct factorisation of x² + 6x + 8? (a) (x − 2)(x − 4) (b) (x + 2)(x + 4) (c) (x + 1)(x + 8) (d) (x + 1)(x + 5)
  21. 21.If a + b = 15 and ab = 50, what is a² + b²? (a) 225 (b) 325 (c) 175 (d) 125
  22. 22.If a + b = 14 and ab = 33, what is a² + b²? (a) 130 (b) 196 (c) 262 (d) 163
  23. 23.If a + b = 7 and ab = 12, what is (a − b)²? (a) 37 (b) 25 (c) 49 (d) 1
  24. 24.If a + b = 7 and ab = 10, what is a² + b²? (a) 49 (b) 39 (c) 29 (d) 69
  25. 25.What is the sum of the roots of x² − 7x + 10 = 0? (a) 6 (b) 10 (c) 7 (d) 8
  26. 26.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 4904 (b) 4624 (c) 5194 (d) 5184
  27. 27.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 72 (c) 71 (d) 73
  28. 28.If a + b = 19 and ab = 84, what is (a − b)²? (a) 5 (b) 361 (c) 193 (d) 25
  29. 29.Which is the correct factorisation of x² + 8x + 12? (a) (x + 2)(x + 6) (b) (x − 2)(x − 6) (c) (x + 1)(x + 12) (d) (x + 1)(x + 7)
  30. 30.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 841 (b) 901 (c) 961 (d) 971
  31. 31.If a + b = 12 and ab = 11, what is a² + b²? (a) 166 (b) 144 (c) 122 (d) 133
  32. 32.If a + b = 19 and ab = 84, what is a² + b²? (a) 361 (b) 277 (c) 193 (d) 529
  33. 33.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x + 1)(x + 15) (c) (x + 7)(x + 9) (d) (x − 7)(x − 9)
  34. 34.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 225 (b) 635 (c) 625 (d) 425
  35. 35.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9035 (b) 9975 (c) 9025 (d) 11025
  36. 36.If a + b = 16 and ab = 55, what is (a − b)²? (a) 36 (b) 6 (c) 146 (d) 256
  37. 37.Which is the correct factorisation of x² + 14x + 48? (a) (x + 6)(x + 8) (b) (x + 1)(x + 13) (c) (x + 1)(x + 48) (d) (x − 6)(x − 8)
  38. 38.If a + b = 8 and ab = 12, what is (a − b)²? (a) 64 (b) 16 (c) 4 (d) 40
  39. 39.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4761 (b) 4899 (c) 5041 (d) 4771
  40. 40.If a + b = 16 and ab = 48, what is (a − b)²? (a) 160 (b) 64 (c) 8 (d) 256

Answer key

  1. (x + 7)(x + 8)
  2. 130
  3. 180
  4. 148
  5. 16
  6. 145
  7. 1156
  8. 17
  9. (x + 2)(x + 3)
  10. (x + 8)(x + 9)
  11. 3136
  12. 1
  13. 1764
  14. 1
  15. 121
  16. 12
  17. 4
  18. 7569
  19. 65
  20. (x + 2)(x + 4)
  21. 125
  22. 130
  23. 1
  24. 29
  25. 7
  26. 5184
  27. 72
  28. 25
  29. (x + 2)(x + 6)
  30. 961
  31. 122
  32. 193
  33. (x + 7)(x + 9)
  34. 625
  35. 9025
  36. 36
  37. (x + 6)(x + 8)
  38. 16
  39. 4761
  40. 64

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