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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 448 · 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.What are the roots of x² − 13x + 36 = 0? (a) 4 and 9 (b) 1 and 12 (c) 1 and 36 (d) 2 and 18
  2. 2.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)
  3. 3.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 4096 (b) 4106 (c) 3136 (d) 3616
  4. 4.If a + b = 19 and ab = 90, what is a² + b²? (a) 181 (b) 271 (c) 361 (d) 541
  5. 5.If a + b = 14 and ab = 24, what is (a − b)²? (a) 148 (b) 196 (c) 10 (d) 100
  6. 6.What is the product of the roots of x² − 7x + 12 = 0? (a) 12 (b) 7 (c) 11 (d) 13
  7. 7.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 11025 (b) 9025 (c) 9975 (d) 9035
  8. 8.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 5184 (b) 4624 (c) 4896 (d) 4634
  9. 9.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2401 (b) 2601 (c) 2611 (d) 2501
  10. 10.If a + b = 14 and ab = 40, what is a² + b²? (a) 276 (b) 196 (c) 116 (d) 156
  11. 11.If a + b = 8 and ab = 7, what is (a − b)²? (a) 50 (b) 6 (c) 36 (d) 64
  12. 12.If a + b = 8 and ab = 7, what is a² + b²? (a) 64 (b) 78 (c) 57 (d) 50
  13. 13.What is the product of the roots of x² − 16x + 63 = 0? (a) 62 (b) 64 (c) 63 (d) 16
  14. 14.If a + b = 14 and ab = 48, what is (a − b)²? (a) 2 (b) 100 (c) 4 (d) 196
  15. 15.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3731 (b) 3721 (c) 3601 (d) 3481
  16. 16.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 14 (c) 15 (d) 16
  17. 17.If a + b = 16 and ab = 55, what is (a − b)²? (a) 146 (b) 256 (c) 6 (d) 36
  18. 18.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 8 and 9 (c) 2 and 36 (d) 1 and 16
  19. 19.If a + b = 9 and ab = 20, what is a² + b²? (a) 121 (b) 41 (c) 61 (d) 81
  20. 20.If a + b = 5 and ab = 4, what is (a − b)²? (a) 17 (b) 9 (c) 3 (d) 25
  21. 21.What is the sum of the roots of x² − 10x + 21 = 0? (a) 9 (b) 21 (c) 10 (d) 11
  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 + 11) (d) (x + 1)(x + 32)
  23. 23.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 1)(x + 14) (c) (x − 7)(x − 8) (d) (x + 7)(x + 8)
  24. 24.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 625 (b) 635 (c) 1225 (d) 875
  25. 25.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 9) (b) (x + 2)(x + 8) (c) (x − 2)(x − 8) (d) (x + 1)(x + 16)
  26. 26.If a + b = 15 and ab = 54, what is (a − b)²? (a) 9 (b) 225 (c) 117 (d) 3
  27. 27.If a + b = 21 and ab = 108, what is a² + b²? (a) 657 (b) 225 (c) 333 (d) 441
  28. 28.If a + b = 19 and ab = 88, what is a² + b²? (a) 273 (b) 185 (c) 537 (d) 361
  29. 29.What is the sum of the roots of x² − 12x + 27 = 0? (a) 13 (b) 11 (c) 12 (d) 27
  30. 30.If a + b = 15 and ab = 44, what is (a − b)²? (a) 137 (b) 7 (c) 49 (d) 225
  31. 31.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 1)(x + 63) (c) (x + 7)(x + 9) (d) (x − 7)(x − 9)
  32. 32.If a + b = 14 and ab = 48, what is a² + b²? (a) 196 (b) 148 (c) 100 (d) 292
  33. 33.What is the product of the roots of x² − 11x + 30 = 0? (a) 31 (b) 11 (c) 29 (d) 30
  34. 34.Which is the correct factorisation of x² + 5x + 6? (a) (x + 2)(x + 3) (b) (x + 1)(x + 6) (c) (x − 2)(x − 3) (d) (x + 1)(x + 4)
  35. 35.If a + b = 21 and ab = 108, what is (a − b)²? (a) 441 (b) 3 (c) 9 (d) 225
  36. 36.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8281 (b) 8291 (c) 7921 (d) 8101
  37. 37.What are the roots of x² − 8x + 15 = 0? (a) 1 and 15 (b) 2 and 6 (c) 3 and 5 (d) 1 and 7
  38. 38.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 3146 (b) 3584 (c) 4096 (d) 3136
  39. 39.If a + b = 12 and ab = 32, what is a² + b²? (a) 112 (b) 208 (c) 80 (d) 144
  40. 40.If a + b = 19 and ab = 84, what is (a − b)²? (a) 25 (b) 361 (c) 5 (d) 193

Answer key

  1. 4 and 9
  2. (x + 3)(x + 5)
  3. 4096
  4. 181
  5. 100
  6. 12
  7. 9025
  8. 4624
  9. 2601
  10. 116
  11. 36
  12. 50
  13. 63
  14. 4
  15. 3721
  16. 15
  17. 36
  18. 8 and 9
  19. 41
  20. 9
  21. 10
  22. (x + 4)(x + 8)
  23. (x + 7)(x + 8)
  24. 625
  25. (x + 2)(x + 8)
  26. 9
  27. 225
  28. 185
  29. 12
  30. 49
  31. (x + 7)(x + 9)
  32. 100
  33. 30
  34. (x + 2)(x + 3)
  35. 9
  36. 8281
  37. 3 and 5
  38. 3136
  39. 80
  40. 25

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