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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 247 · 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 + 36) (b) (x + 1)(x + 12) (c) (x + 4)(x + 9) (d) (x − 4)(x − 9)
  2. 2.If a + b = 11 and ab = 28, what is a² + b²? (a) 121 (b) 93 (c) 177 (d) 65
  3. 3.Which is the correct factorisation of x² + 5x + 6? (a) (x − 2)(x − 3) (b) (x + 1)(x + 4) (c) (x + 1)(x + 6) (d) (x + 2)(x + 3)
  4. 4.What is the product of the roots of x² − 17x + 72 = 0? (a) 72 (b) 71 (c) 17 (d) 73
  5. 5.Which is the correct factorisation of x² + 12x + 27? (a) (x + 3)(x + 9) (b) (x + 1)(x + 27) (c) (x − 3)(x − 9) (d) (x + 1)(x + 11)
  6. 6.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 676 (b) 1156 (c) 916 (d) 1166
  7. 7.If a + b = 8 and ab = 12, what is a² + b²? (a) 64 (b) 88 (c) 52 (d) 40
  8. 8.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 8464 (b) 7754 (c) 8096 (d) 7744
  9. 9.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 4634 (b) 5184 (c) 4624 (d) 4896
  10. 10.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 7)(x + 8) (c) (x + 1)(x + 56) (d) (x + 1)(x + 14)
  11. 11.Which is the correct factorisation of x² + 12x + 32? (a) (x + 4)(x + 8) (b) (x − 4)(x − 8) (c) (x + 1)(x + 11) (d) (x + 1)(x + 32)
  12. 12.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 7225 (b) 8125 (c) 9035 (d) 9025
  13. 13.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 441 (b) 399 (c) 361 (d) 371
  14. 14.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6084 (b) 6724 (c) 6404 (d) 6734
  15. 15.If a + b = 16 and ab = 60, what is a² + b²? (a) 196 (b) 376 (c) 136 (d) 256
  16. 16.What is the sum of the roots of x² − 12x + 35 = 0? (a) 12 (b) 13 (c) 11 (d) 35
  17. 17.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)
  18. 18.If a + b = 20 and ab = 99, what is (a − b)²? (a) 202 (b) 400 (c) 4 (d) 2
  19. 19.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4884 (b) 5476 (c) 4356 (d) 4366
  20. 20.If a + b = 17 and ab = 72, what is a² + b²? (a) 145 (b) 217 (c) 289 (d) 433
  21. 21.If a + b = 18 and ab = 80, what is (a − b)²? (a) 2 (b) 164 (c) 324 (d) 4
  22. 22.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 15 (c) 16 (d) 14
  23. 23.If a + b = 10 and ab = 16, what is a² + b²? (a) 84 (b) 68 (c) 100 (d) 132
  24. 24.What is the sum of the roots of x² − 10x + 24 = 0? (a) 11 (b) 9 (c) 10 (d) 24
  25. 25.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1849 (b) 1591 (c) 1379 (d) 1369
  26. 26.What are the roots of x² − 17x + 72 = 0? (a) 2 and 36 (b) 1 and 16 (c) 8 and 9 (d) 1 and 72
  27. 27.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
  28. 28.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3481 (b) 3731 (c) 3601 (d) 3721
  29. 29.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 266 (b) 384 (c) 576 (d) 256
  30. 30.If a + b = 5 and ab = 6, what is (a − b)²? (a) 19 (b) 25 (c) 1 (d) 13
  31. 31.What is the sum of the roots of x² − 13x + 40 = 0? (a) 13 (b) 14 (c) 12 (d) 40
  32. 32.What is the product of the roots of x² − 15x + 56 = 0? (a) 15 (b) 57 (c) 56 (d) 55
  33. 33.If a + b = 16 and ab = 63, what is a² + b²? (a) 382 (b) 193 (c) 256 (d) 130
  34. 34.What is the sum of the roots of x² − 11x + 28 = 0? (a) 12 (b) 10 (c) 28 (d) 11
  35. 35.What is the product of the roots of x² − 9x + 18 = 0? (a) 18 (b) 9 (c) 19 (d) 17
  36. 36.If a + b = 20 and ab = 99, what is a² + b²? (a) 598 (b) 400 (c) 301 (d) 202
  37. 37.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1599 (b) 1681 (c) 1531 (d) 1521
  38. 38.Which is the correct factorisation of x² + 15x + 54? (a) (x + 6)(x + 9) (b) (x + 1)(x + 14) (c) (x − 6)(x − 9) (d) (x + 1)(x + 54)
  39. 39.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 10) (b) (x − 3)(x − 8) (c) (x + 3)(x + 8) (d) (x + 1)(x + 24)
  40. 40.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 40) (b) (x + 1)(x + 12) (c) (x − 5)(x − 8) (d) (x + 5)(x + 8)

Answer key

  1. (x + 4)(x + 9)
  2. 65
  3. (x + 2)(x + 3)
  4. 72
  5. (x + 3)(x + 9)
  6. 1156
  7. 40
  8. 7744
  9. 4624
  10. (x + 7)(x + 8)
  11. (x + 4)(x + 8)
  12. 9025
  13. 361
  14. 6724
  15. 136
  16. 12
  17. (x + 6)(x + 8)
  18. 4
  19. 4356
  20. 145
  21. 4
  22. 15
  23. 68
  24. 10
  25. 1369
  26. 8 and 9
  27. 3 and 5
  28. 3721
  29. 256
  30. 1
  31. 13
  32. 56
  33. 130
  34. 11
  35. 18
  36. 202
  37. 1521
  38. (x + 6)(x + 9)
  39. (x + 3)(x + 8)
  40. (x + 5)(x + 8)

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