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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 305 · 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.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 4875 (b) 4225 (c) 5625 (d) 4235
  2. 2.If a + b = 18 and ab = 80, what is (a − b)²? (a) 164 (b) 2 (c) 324 (d) 4
  3. 3.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)
  4. 4.What is the sum of the roots of x² − 8x + 12 = 0? (a) 8 (b) 9 (c) 12 (d) 7
  5. 5.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 35) (b) (x + 5)(x + 7) (c) (x − 5)(x − 7) (d) (x + 1)(x + 11)
  6. 6.What are the roots of x² − 17x + 72 = 0? (a) 2 and 36 (b) 1 and 72 (c) 8 and 9 (d) 1 and 16
  7. 7.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2401 (b) 2411 (c) 2499 (d) 2601
  8. 8.If a + b = 10 and ab = 24, what is a² + b²? (a) 76 (b) 148 (c) 100 (d) 52
  9. 9.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6724 (b) 6084 (c) 6396 (d) 6094
  10. 10.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 8084 (b) 8836 (c) 7406 (d) 7396
  11. 11.What are the roots of x² − 13x + 36 = 0? (a) 1 and 36 (b) 1 and 12 (c) 2 and 18 (d) 4 and 9
  12. 12.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 4356 (b) 5476 (c) 5486 (d) 4916
  13. 13.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 4896 (b) 5184 (c) 4624 (d) 4634
  14. 14.If a + b = 15 and ab = 56, what is (a − b)²? (a) 113 (b) 169 (c) 225 (d) 1
  15. 15.What is the product of the roots of x² − 7x + 12 = 0? (a) 7 (b) 13 (c) 11 (d) 12
  16. 16.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 71 (c) 17 (d) 72
  17. 17.Which is the correct factorisation of x² + 6x + 8? (a) (x − 2)(x − 4) (b) (x + 2)(x + 4) (c) (x + 1)(x + 8) (d) (x + 1)(x + 5)
  18. 18.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 784 (b) 1024 (c) 896 (d) 794
  19. 19.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 14 (c) 15 (d) 16
  20. 20.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3721 (b) 3481 (c) 3491 (d) 3599
  21. 21.What are the roots of x² − 12x + 32 = 0? (a) 1 and 11 (b) 4 and 8 (c) 2 and 16 (d) 1 and 32
  22. 22.If a + b = 19 and ab = 90, what is (a − b)²? (a) 181 (b) 1 (c) 271 (d) 361
  23. 23.If a + b = 17 and ab = 66, what is a² + b²? (a) 289 (b) 157 (c) 421 (d) 223
  24. 24.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1099 (b) 909 (c) 1089 (d) 729
  25. 25.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x + 1)(x + 72) (c) (x − 8)(x − 9) (d) (x + 1)(x + 16)
  26. 26.If a + b = 15 and ab = 36, what is a² + b²? (a) 189 (b) 153 (c) 225 (d) 297
  27. 27.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)
  28. 28.What is the sum of the roots of x² − 8x + 15 = 0? (a) 15 (b) 9 (c) 7 (d) 8
  29. 29.If a + b = 13 and ab = 36, what is (a − b)²? (a) 25 (b) 169 (c) 97 (d) 5
  30. 30.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x + 1)(x + 12) (c) (x + 6)(x + 7) (d) (x − 6)(x − 7)
  31. 31.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2809 (b) 2209 (c) 2509 (d) 2819
  32. 32.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 256 (b) 586 (c) 576 (d) 416
  33. 33.What is the sum of the roots of x² − 10x + 21 = 0? (a) 9 (b) 10 (c) 21 (d) 11
  34. 34.If a + b = 3 and ab = 2, what is a² + b²? (a) 5 (b) 13 (c) 9 (d) 7
  35. 35.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 904 (b) 1034 (c) 1024 (d) 784
  36. 36.If a + b = 17 and ab = 72, what is a² + b²? (a) 433 (b) 289 (c) 217 (d) 145
  37. 37.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 8125 (b) 7225 (c) 9035 (d) 9025
  38. 38.Which is the correct factorisation of x² + 9x + 18? (a) (x + 3)(x + 6) (b) (x + 1)(x + 18) (c) (x + 1)(x + 8) (d) (x − 3)(x − 6)
  39. 39.If a + b = 8 and ab = 15, what is (a − b)²? (a) 34 (b) 64 (c) 4 (d) 2
  40. 40.What are the roots of x² − 13x + 40 = 0? (a) 1 and 40 (b) 2 and 20 (c) 5 and 8 (d) 1 and 12

Answer key

  1. 4225
  2. 4
  3. (x + 6)(x + 8)
  4. 8
  5. (x + 5)(x + 7)
  6. 8 and 9
  7. 2401
  8. 52
  9. 6084
  10. 7396
  11. 4 and 9
  12. 5476
  13. 4624
  14. 1
  15. 12
  16. 72
  17. (x + 2)(x + 4)
  18. 784
  19. 15
  20. 3481
  21. 4 and 8
  22. 1
  23. 157
  24. 1089
  25. (x + 8)(x + 9)
  26. 153
  27. (x + 7)(x + 9)
  28. 8
  29. 25
  30. (x + 6)(x + 7)
  31. 2809
  32. 576
  33. 10
  34. 5
  35. 1024
  36. 145
  37. 9025
  38. (x + 3)(x + 6)
  39. 4
  40. 5 and 8

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