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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 420 · 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 73²? (a) 4489 (b) 5339 (c) 5329 (d) 4909
  2. 2.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 1156 (b) 676 (c) 916 (d) 1166
  3. 3.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 1235 (b) 1575 (c) 2025 (d) 1225
  4. 4.If a + b = 15 and ab = 56, what is a² + b²? (a) 225 (b) 169 (c) 337 (d) 113
  5. 5.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 10) (b) (x − 2)(x − 9) (c) (x + 2)(x + 9) (d) (x + 1)(x + 18)
  6. 6.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 361 (b) 441 (c) 399 (d) 371
  7. 7.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 1625 (b) 1225 (c) 2025 (d) 2035
  8. 8.If a + b = 10 and ab = 9, what is (a − b)²? (a) 8 (b) 100 (c) 64 (d) 82
  9. 9.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x − 6)(x − 7) (c) (x + 1)(x + 42) (d) (x + 6)(x + 7)
  10. 10.What is the product of the roots of x² − 14x + 45 = 0? (a) 44 (b) 14 (c) 45 (d) 46
  11. 11.If a + b = 16 and ab = 48, what is a² + b²? (a) 160 (b) 256 (c) 208 (d) 352
  12. 12.If a + b = 15 and ab = 56, what is (a − b)²? (a) 113 (b) 169 (c) 225 (d) 1
  13. 13.If a + b = 9 and ab = 20, what is (a − b)²? (a) 81 (b) 61 (c) 1 (d) 41
  14. 14.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 901 (b) 961 (c) 971 (d) 841
  15. 15.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 9025 (b) 9035 (c) 7225 (d) 8125
  16. 16.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 7)(x + 8) (c) (x − 7)(x − 8) (d) (x + 1)(x + 14)
  17. 17.If a + b = 11 and ab = 28, what is (a − b)²? (a) 121 (b) 65 (c) 3 (d) 9
  18. 18.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6084 (b) 6724 (c) 6734 (d) 6404
  19. 19.If a + b = 15 and ab = 44, what is a² + b²? (a) 313 (b) 225 (c) 181 (d) 137
  20. 20.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 11025 (b) 10025 (c) 9025 (d) 11035
  21. 21.Which is the correct factorisation of x² + 9x + 20? (a) (x + 4)(x + 5) (b) (x + 1)(x + 8) (c) (x + 1)(x + 20) (d) (x − 4)(x − 5)
  22. 22.If a + b = 15 and ab = 54, what is a² + b²? (a) 333 (b) 117 (c) 171 (d) 225
  23. 23.If a + b = 18 and ab = 77, what is a² + b²? (a) 478 (b) 247 (c) 170 (d) 324
  24. 24.Which is the correct factorisation of x² + 7x + 10? (a) (x − 2)(x − 5) (b) (x + 2)(x + 5) (c) (x + 1)(x + 6) (d) (x + 1)(x + 10)
  25. 25.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 71 (c) 73 (d) 72
  26. 26.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 12) (b) (x + 5)(x + 8) (c) (x − 5)(x − 8) (d) (x + 1)(x + 40)
  27. 27.What is the product of the roots of x² − 12x + 27 = 0? (a) 28 (b) 27 (c) 26 (d) 12
  28. 28.If a + b = 11 and ab = 30, what is (a − b)²? (a) 91 (b) 1 (c) 61 (d) 121
  29. 29.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 8099 (b) 8281 (c) 7931 (d) 7921
  30. 30.What is the sum of the roots of x² − 6x + 8 = 0? (a) 7 (b) 5 (c) 8 (d) 6
  31. 31.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9996 (b) 9604 (c) 10404 (d) 9614
  32. 32.If a + b = 8 and ab = 12, what is (a − b)²? (a) 64 (b) 40 (c) 16 (d) 4
  33. 33.What is the product of the roots of x² − 12x + 32 = 0? (a) 31 (b) 33 (c) 32 (d) 12
  34. 34.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 5194 (b) 4904 (c) 5184 (d) 4624
  35. 35.What is the sum of the roots of x² − 7x + 10 = 0? (a) 6 (b) 8 (c) 10 (d) 7
  36. 36.If a + b = 6 and ab = 8, what is a² + b²? (a) 52 (b) 28 (c) 20 (d) 36
  37. 37.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 17 (c) 18 (d) 16
  38. 38.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 16) (c) (x + 1)(x + 72) (d) (x + 8)(x + 9)
  39. 39.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 625 (b) 635 (c) 225 (d) 425
  40. 40.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3604 (b) 3854 (c) 3844 (d) 3364

Answer key

  1. 5329
  2. 1156
  3. 1225
  4. 113
  5. (x + 2)(x + 9)
  6. 361
  7. 2025
  8. 64
  9. (x + 6)(x + 7)
  10. 45
  11. 160
  12. 1
  13. 1
  14. 961
  15. 9025
  16. (x + 7)(x + 8)
  17. 9
  18. 6724
  19. 137
  20. 11025
  21. (x + 4)(x + 5)
  22. 117
  23. 170
  24. (x + 2)(x + 5)
  25. 72
  26. (x + 5)(x + 8)
  27. 27
  28. 1
  29. 7921
  30. 6
  31. 9604
  32. 16
  33. 32
  34. 5184
  35. 7
  36. 20
  37. 17
  38. (x + 8)(x + 9)
  39. 625
  40. 3844

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