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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 859 · 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.If a + b = 19 and ab = 88, what is a² + b²? (a) 273 (b) 185 (c) 537 (d) 361
  2. 2.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6724 (b) 6094 (c) 6084 (d) 6396
  3. 3.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x + 1)(x + 15) (c) (x − 7)(x − 9) (d) (x + 1)(x + 63)
  4. 4.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x + 1)(x + 16) (c) (x − 8)(x − 9) (d) (x + 1)(x + 72)
  5. 5.Which is the correct factorisation of x² + 13x + 42? (a) (x + 6)(x + 7) (b) (x − 6)(x − 7) (c) (x + 1)(x + 12) (d) (x + 1)(x + 42)
  6. 6.What is the sum of the roots of x² − 11x + 30 = 0? (a) 12 (b) 11 (c) 30 (d) 10
  7. 7.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 7579 (b) 7569 (c) 8091 (d) 8649
  8. 8.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 5486 (b) 4356 (c) 5476 (d) 4916
  9. 9.What is the product of the roots of x² − 11x + 18 = 0? (a) 11 (b) 19 (c) 17 (d) 18
  10. 10.If a + b = 7 and ab = 12, what is a² + b²? (a) 49 (b) 37 (c) 25 (d) 73
  11. 11.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3604 (b) 3364 (c) 3854 (d) 3844
  12. 12.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 18 (c) 17 (d) 16
  13. 13.If a + b = 18 and ab = 72, what is (a − b)²? (a) 180 (b) 324 (c) 6 (d) 36
  14. 14.What is the sum of the roots of x² − 6x + 8 = 0? (a) 5 (b) 6 (c) 8 (d) 7
  15. 15.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 36) (b) (x − 4)(x − 9) (c) (x + 1)(x + 12) (d) (x + 4)(x + 9)
  16. 16.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 576 (b) 586 (c) 416 (d) 256
  17. 17.What are the roots of x² − 8x + 12 = 0? (a) 1 and 7 (b) 1 and 12 (c) 2 and 6 (d) 3 and 4
  18. 18.If a + b = 16 and ab = 48, what is a² + b²? (a) 160 (b) 352 (c) 208 (d) 256
  19. 19.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 4624 (b) 4896 (c) 4634 (d) 5184
  20. 20.What are the roots of x² − 8x + 15 = 0? (a) 1 and 15 (b) 1 and 7 (c) 3 and 5 (d) 2 and 6
  21. 21.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 6)(x + 8) (c) (x + 1)(x + 13) (d) (x + 1)(x + 48)
  22. 22.What is the product of the roots of x² − 14x + 48 = 0? (a) 48 (b) 47 (c) 14 (d) 49
  23. 23.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 5625 (b) 6425 (c) 7225 (d) 7235
  24. 24.If a + b = 17 and ab = 72, what is a² + b²? (a) 289 (b) 217 (c) 433 (d) 145
  25. 25.What is the product of the roots of x² − 10x + 21 = 0? (a) 20 (b) 22 (c) 21 (d) 10
  26. 26.If a + b = 17 and ab = 72, what is (a − b)²? (a) 217 (b) 1 (c) 289 (d) 145
  27. 27.If a + b = 21 and ab = 108, what is a² + b²? (a) 333 (b) 657 (c) 441 (d) 225
  28. 28.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 6391 (b) 6889 (c) 5929 (d) 5939
  29. 29.What is the sum of the roots of x² − 8x + 15 = 0? (a) 9 (b) 8 (c) 7 (d) 15
  30. 30.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10619 (b) 10609 (c) 9409 (d) 10009
  31. 31.If a + b = 17 and ab = 66, what is a² + b²? (a) 157 (b) 421 (c) 289 (d) 223
  32. 32.What are the roots of x² − 11x + 18 = 0? (a) 1 and 18 (b) 3 and 6 (c) 2 and 9 (d) 1 and 10
  33. 33.If a + b = 10 and ab = 24, what is a² + b²? (a) 148 (b) 100 (c) 76 (d) 52
  34. 34.If a + b = 19 and ab = 88, what is (a − b)²? (a) 185 (b) 3 (c) 361 (d) 9
  35. 35.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 5184 (b) 4904 (c) 5194 (d) 4624
  36. 36.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 396 (b) 484 (c) 334 (d) 324
  37. 37.If a + b = 14 and ab = 33, what is a² + b²? (a) 196 (b) 163 (c) 130 (d) 262
  38. 38.If a + b = 11 and ab = 24, what is a² + b²? (a) 169 (b) 121 (c) 73 (d) 97
  39. 39.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 32) (b) (x − 4)(x − 8) (c) (x + 1)(x + 11) (d) (x + 4)(x + 8)
  40. 40.What are the roots of x² − 10x + 21 = 0? (a) 1 and 21 (b) 1 and 9 (c) 2 and 8 (d) 3 and 7

Answer key

  1. 185
  2. 6084
  3. (x + 7)(x + 9)
  4. (x + 8)(x + 9)
  5. (x + 6)(x + 7)
  6. 11
  7. 7569
  8. 5476
  9. 18
  10. 25
  11. 3844
  12. 17
  13. 36
  14. 6
  15. (x + 4)(x + 9)
  16. 576
  17. 2 and 6
  18. 160
  19. 4624
  20. 3 and 5
  21. (x + 6)(x + 8)
  22. 48
  23. 7225
  24. 145
  25. 21
  26. 1
  27. 225
  28. 5929
  29. 8
  30. 10609
  31. 157
  32. 2 and 9
  33. 52
  34. 9
  35. 5184
  36. 324
  37. 130
  38. 73
  39. (x + 4)(x + 8)
  40. 3 and 7

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