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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 757 · 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 + 1)(x + 14) (c) (x + 1)(x + 56) (d) (x − 7)(x − 8)
  2. 2.What is the sum of the roots of x² − 12x + 32 = 0? (a) 13 (b) 11 (c) 32 (d) 12
  3. 3.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 2525 (b) 2025 (c) 3035 (d) 3025
  4. 4.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1599 (b) 1531 (c) 1681 (d) 1521
  5. 5.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 1)(x + 48) (c) (x + 6)(x + 8) (d) (x + 1)(x + 13)
  6. 6.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2401 (b) 2499 (c) 2411 (d) 2601
  7. 7.If a + b = 21 and ab = 108, what is (a − b)²? (a) 9 (b) 441 (c) 3 (d) 225
  8. 8.What are the roots of x² − 17x + 72 = 0? (a) 8 and 9 (b) 1 and 16 (c) 2 and 36 (d) 1 and 72
  9. 9.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 384 (b) 266 (c) 256 (d) 576
  10. 10.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 9) (b) (x − 4)(x − 6) (c) (x + 4)(x + 6) (d) (x + 1)(x + 24)
  11. 11.If a + b = 18 and ab = 72, what is a² + b²? (a) 324 (b) 468 (c) 180 (d) 252
  12. 12.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 7)(x + 9) (c) (x − 7)(x − 9) (d) (x + 1)(x + 63)
  13. 13.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 12) (b) (x + 1)(x + 6) (c) (x − 3)(x − 4) (d) (x + 3)(x + 4)
  14. 14.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)
  15. 15.If a + b = 13 and ab = 40, what is a² + b²? (a) 89 (b) 169 (c) 129 (d) 249
  16. 16.Which is the correct factorisation of x² + 5x + 6? (a) (x − 2)(x − 3) (b) (x + 2)(x + 3) (c) (x + 1)(x + 4) (d) (x + 1)(x + 6)
  17. 17.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 2475 (b) 3025 (c) 2025 (d) 2035
  18. 18.If a + b = 20 and ab = 96, what is a² + b²? (a) 400 (b) 208 (c) 304 (d) 592
  19. 19.If a + b = 9 and ab = 14, what is a² + b²? (a) 67 (b) 81 (c) 53 (d) 109
  20. 20.What are the roots of x² − 13x + 42 = 0? (a) 2 and 21 (b) 6 and 7 (c) 1 and 42 (d) 1 and 12
  21. 21.If a + b = 14 and ab = 48, what is (a − b)²? (a) 4 (b) 196 (c) 2 (d) 100
  22. 22.Which is the correct factorisation of x² + 12x + 32? (a) (x − 4)(x − 8) (b) (x + 4)(x + 8) (c) (x + 1)(x + 32) (d) (x + 1)(x + 11)
  23. 23.If a + b = 16 and ab = 63, what is (a − b)²? (a) 2 (b) 4 (c) 130 (d) 256
  24. 24.If a + b = 9 and ab = 8, what is a² + b²? (a) 97 (b) 81 (c) 73 (d) 65
  25. 25.If a + b = 14 and ab = 40, what is (a − b)²? (a) 196 (b) 6 (c) 116 (d) 36
  26. 26.What is the product of the roots of x² − 14x + 45 = 0? (a) 14 (b) 46 (c) 45 (d) 44
  27. 27.If a + b = 14 and ab = 45, what is (a − b)²? (a) 4 (b) 106 (c) 196 (d) 16
  28. 28.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 1)(x + 16) (c) (x + 8)(x + 9) (d) (x − 8)(x − 9)
  29. 29.Which is the correct factorisation of x² + 8x + 15? (a) (x + 3)(x + 5) (b) (x + 1)(x + 15) (c) (x − 3)(x − 5) (d) (x + 1)(x + 7)
  30. 30.If a + b = 17 and ab = 60, what is a² + b²? (a) 409 (b) 229 (c) 289 (d) 169
  31. 31.If a + b = 19 and ab = 90, what is a² + b²? (a) 271 (b) 181 (c) 361 (d) 541
  32. 32.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 8) (b) (x + 2)(x + 7) (c) (x − 2)(x − 7) (d) (x + 1)(x + 14)
  33. 33.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 3584 (b) 3146 (c) 3136 (d) 4096
  34. 34.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8291 (b) 8281 (c) 7921 (d) 8101
  35. 35.Which is the correct factorisation of x² + 14x + 45? (a) (x + 5)(x + 9) (b) (x + 1)(x + 13) (c) (x + 1)(x + 45) (d) (x − 5)(x − 9)
  36. 36.What are the roots of x² − 12x + 27 = 0? (a) 1 and 27 (b) 3 and 9 (c) 2 and 10 (d) 1 and 11
  37. 37.If a + b = 11 and ab = 18, what is (a − b)²? (a) 121 (b) 7 (c) 85 (d) 49
  38. 38.If a + b = 6 and ab = 5, what is (a − b)²? (a) 16 (b) 36 (c) 26 (d) 4
  39. 39.What is the sum of the roots of x² − 8x + 15 = 0? (a) 9 (b) 8 (c) 7 (d) 15
  40. 40.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 324 (b) 404 (c) 494 (d) 484

Answer key

  1. (x + 7)(x + 8)
  2. 12
  3. 3025
  4. 1521
  5. (x + 6)(x + 8)
  6. 2401
  7. 9
  8. 8 and 9
  9. 256
  10. (x + 4)(x + 6)
  11. 180
  12. (x + 7)(x + 9)
  13. (x + 3)(x + 4)
  14. (x + 2)(x + 8)
  15. 89
  16. (x + 2)(x + 3)
  17. 2025
  18. 208
  19. 53
  20. 6 and 7
  21. 4
  22. (x + 4)(x + 8)
  23. 4
  24. 65
  25. 36
  26. 45
  27. 16
  28. (x + 8)(x + 9)
  29. (x + 3)(x + 5)
  30. 169
  31. 181
  32. (x + 2)(x + 7)
  33. 3136
  34. 8281
  35. (x + 5)(x + 9)
  36. 3 and 9
  37. 49
  38. 16
  39. 8
  40. 484

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