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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 345 · 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 + 17x + 72? (a) (x + 1)(x + 16) (b) (x − 8)(x − 9) (c) (x + 8)(x + 9) (d) (x + 1)(x + 72)
  2. 2.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x + 7)(x + 8) (c) (x − 7)(x − 8) (d) (x + 1)(x + 56)
  3. 3.If a + b = 12 and ab = 35, what is (a − b)²? (a) 4 (b) 74 (c) 144 (d) 2
  4. 4.If a + b = 16 and ab = 63, what is (a − b)²? (a) 256 (b) 2 (c) 130 (d) 4
  5. 5.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 409 (b) 529 (c) 539 (d) 289
  6. 6.If a + b = 18 and ab = 77, what is (a − b)²? (a) 170 (b) 16 (c) 324 (d) 4
  7. 7.If a + b = 12 and ab = 35, what is a² + b²? (a) 214 (b) 109 (c) 74 (d) 144
  8. 8.If a + b = 18 and ab = 80, what is a² + b²? (a) 484 (b) 164 (c) 324 (d) 244
  9. 9.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 18) (b) (x − 2)(x − 9) (c) (x + 1)(x + 10) (d) (x + 2)(x + 9)
  10. 10.If a + b = 7 and ab = 10, what is (a − b)²? (a) 29 (b) 9 (c) 3 (d) 49
  11. 11.If a + b = 13 and ab = 36, what is (a − b)²? (a) 5 (b) 25 (c) 97 (d) 169
  12. 12.If a + b = 11 and ab = 24, what is (a − b)²? (a) 5 (b) 121 (c) 25 (d) 73
  13. 13.If a + b = 18 and ab = 77, what is a² + b²? (a) 478 (b) 324 (c) 170 (d) 247
  14. 14.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6241 (b) 6251 (c) 6399 (d) 6561
  15. 15.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 4916 (b) 4356 (c) 5486 (d) 5476
  16. 16.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 35) (b) (x + 1)(x + 11) (c) (x − 5)(x − 7) (d) (x + 5)(x + 7)
  17. 17.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)
  18. 18.If a + b = 20 and ab = 96, what is a² + b²? (a) 592 (b) 400 (c) 304 (d) 208
  19. 19.What are the roots of x² − 12x + 27 = 0? (a) 1 and 27 (b) 2 and 10 (c) 3 and 9 (d) 1 and 11
  20. 20.If a + b = 5 and ab = 4, what is a² + b²? (a) 33 (b) 25 (c) 17 (d) 21
  21. 21.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 1235 (b) 1575 (c) 1225 (d) 2025
  22. 22.If a + b = 8 and ab = 12, what is a² + b²? (a) 52 (b) 64 (c) 40 (d) 88
  23. 23.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2916 (b) 2126 (c) 2116 (d) 2484
  24. 24.What is the sum of the roots of x² − 9x + 18 = 0? (a) 10 (b) 8 (c) 9 (d) 18
  25. 25.If a + b = 13 and ab = 40, what is a² + b²? (a) 129 (b) 169 (c) 249 (d) 89
  26. 26.If a + b = 9 and ab = 8, what is a² + b²? (a) 73 (b) 97 (c) 81 (d) 65
  27. 27.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6724 (b) 6084 (c) 6734 (d) 6404
  28. 28.What is the sum of the roots of x² − 13x + 42 = 0? (a) 42 (b) 12 (c) 13 (d) 14
  29. 29.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3721 (b) 3491 (c) 3481 (d) 3599
  30. 30.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2491 (b) 2219 (c) 2809 (d) 2209
  31. 31.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10001 (b) 9801 (c) 10211 (d) 10201
  32. 32.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 13) (b) (x − 5)(x − 9) (c) (x + 1)(x + 45) (d) (x + 5)(x + 9)
  33. 33.What is the sum of the roots of x² − 15x + 54 = 0? (a) 15 (b) 54 (c) 16 (d) 14
  34. 34.What is the product of the roots of x² − 16x + 63 = 0? (a) 64 (b) 63 (c) 16 (d) 62
  35. 35.If a + b = 17 and ab = 70, what is (a − b)²? (a) 9 (b) 149 (c) 3 (d) 289
  36. 36.If a + b = 14 and ab = 45, what is (a − b)²? (a) 196 (b) 16 (c) 106 (d) 4
  37. 37.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 9) (b) (x − 4)(x − 6) (c) (x + 1)(x + 24) (d) (x + 4)(x + 6)
  38. 38.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 12) (b) (x + 3)(x + 4) (c) (x + 1)(x + 6) (d) (x − 3)(x − 4)
  39. 39.If a + b = 8 and ab = 7, what is a² + b²? (a) 78 (b) 57 (c) 50 (d) 64
  40. 40.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1531 (b) 1521 (c) 1681 (d) 1599

Answer key

  1. (x + 8)(x + 9)
  2. (x + 7)(x + 8)
  3. 4
  4. 4
  5. 529
  6. 16
  7. 74
  8. 164
  9. (x + 2)(x + 9)
  10. 9
  11. 25
  12. 25
  13. 170
  14. 6241
  15. 5476
  16. (x + 5)(x + 7)
  17. (x + 2)(x + 3)
  18. 208
  19. 3 and 9
  20. 17
  21. 1225
  22. 40
  23. 2116
  24. 9
  25. 89
  26. 65
  27. 6724
  28. 13
  29. 3481
  30. 2209
  31. 10201
  32. (x + 5)(x + 9)
  33. 15
  34. 63
  35. 9
  36. 16
  37. (x + 4)(x + 6)
  38. (x + 3)(x + 4)
  39. 50
  40. 1521

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