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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 627 · 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 is the product of the roots of x² − 6x + 8 = 0? (a) 9 (b) 7 (c) 6 (d) 8
  2. 2.What is the product of the roots of x² − 11x + 18 = 0? (a) 18 (b) 11 (c) 17 (d) 19
  3. 3.If a + b = 8 and ab = 7, what is (a − b)²? (a) 50 (b) 36 (c) 64 (d) 6
  4. 4.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2501 (b) 2611 (c) 2601 (d) 2401
  5. 5.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 1)(x + 42) (c) (x + 6)(x + 7) (d) (x + 1)(x + 12)
  6. 6.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6084 (b) 6724 (c) 6404 (d) 6734
  7. 7.What is the product of the roots of x² − 9x + 18 = 0? (a) 19 (b) 9 (c) 18 (d) 17
  8. 8.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 875 (b) 635 (c) 625 (d) 1225
  9. 9.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 6) (b) (x + 2)(x + 3) (c) (x + 1)(x + 4) (d) (x − 2)(x − 3)
  10. 10.Which is the correct factorisation of x² + 6x + 8? (a) (x − 2)(x − 4) (b) (x + 1)(x + 8) (c) (x + 2)(x + 4) (d) (x + 1)(x + 5)
  11. 11.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 3616 (b) 3136 (c) 4096 (d) 4106
  12. 12.If a + b = 18 and ab = 77, what is a² + b²? (a) 324 (b) 170 (c) 247 (d) 478
  13. 13.Which is the correct factorisation of x² + 8x + 12? (a) (x + 2)(x + 6) (b) (x − 2)(x − 6) (c) (x + 1)(x + 12) (d) (x + 1)(x + 7)
  14. 14.If a + b = 13 and ab = 30, what is a² + b²? (a) 109 (b) 229 (c) 139 (d) 169
  15. 15.If a + b = 19 and ab = 90, what is (a − b)²? (a) 361 (b) 181 (c) 1 (d) 271
  16. 16.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 10816 (b) 9216 (c) 10016 (d) 10826
  17. 17.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 2025 (b) 2525 (c) 3035 (d) 3025
  18. 18.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 8)(x + 9) (c) (x − 8)(x − 9) (d) (x + 1)(x + 72)
  19. 19.What is the product of the roots of x² − 15x + 56 = 0? (a) 15 (b) 55 (c) 57 (d) 56
  20. 20.What is the sum of the roots of x² − 11x + 30 = 0? (a) 12 (b) 11 (c) 10 (d) 30
  21. 21.If a + b = 13 and ab = 42, what is a² + b²? (a) 253 (b) 169 (c) 127 (d) 85
  22. 22.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1764 (b) 1604 (c) 1774 (d) 1444
  23. 23.What is the product of the roots of x² − 15x + 54 = 0? (a) 15 (b) 55 (c) 54 (d) 53
  24. 24.If a + b = 15 and ab = 44, what is a² + b²? (a) 181 (b) 313 (c) 225 (d) 137
  25. 25.Which is the correct factorisation of x² + 9x + 18? (a) (x − 3)(x − 6) (b) (x + 1)(x + 18) (c) (x + 3)(x + 6) (d) (x + 1)(x + 8)
  26. 26.If a + b = 16 and ab = 48, what is a² + b²? (a) 352 (b) 256 (c) 160 (d) 208
  27. 27.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x + 1)(x + 56) (c) (x + 7)(x + 8) (d) (x − 7)(x − 8)
  28. 28.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 5041 (b) 4771 (c) 4899 (d) 4761
  29. 29.What is the sum of the roots of x² − 10x + 24 = 0? (a) 10 (b) 11 (c) 9 (d) 24
  30. 30.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 7235 (b) 9025 (c) 7225 (d) 8075
  31. 31.What is the sum of the roots of x² − 13x + 42 = 0? (a) 13 (b) 14 (c) 42 (d) 12
  32. 32.If a + b = 14 and ab = 45, what is (a − b)²? (a) 196 (b) 16 (c) 4 (d) 106
  33. 33.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 896 (b) 1024 (c) 794 (d) 784
  34. 34.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 9025 (b) 11025 (c) 11035 (d) 10025
  35. 35.If a + b = 20 and ab = 99, what is (a − b)²? (a) 400 (b) 2 (c) 202 (d) 4
  36. 36.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 6) (b) (x − 2)(x − 5) (c) (x + 1)(x + 10) (d) (x + 2)(x + 5)
  37. 37.If a + b = 19 and ab = 90, what is a² + b²? (a) 271 (b) 181 (c) 541 (d) 361
  38. 38.What is the sum of the roots of x² − 7x + 10 = 0? (a) 7 (b) 6 (c) 8 (d) 10
  39. 39.What is the product of the roots of x² − 14x + 48 = 0? (a) 49 (b) 47 (c) 48 (d) 14
  40. 40.If a + b = 17 and ab = 66, what is (a − b)²? (a) 25 (b) 5 (c) 289 (d) 157

Answer key

  1. 8
  2. 18
  3. 36
  4. 2601
  5. (x + 6)(x + 7)
  6. 6724
  7. 18
  8. 625
  9. (x + 2)(x + 3)
  10. (x + 2)(x + 4)
  11. 4096
  12. 170
  13. (x + 2)(x + 6)
  14. 109
  15. 1
  16. 10816
  17. 3025
  18. (x + 8)(x + 9)
  19. 56
  20. 11
  21. 85
  22. 1764
  23. 54
  24. 137
  25. (x + 3)(x + 6)
  26. 160
  27. (x + 7)(x + 8)
  28. 4761
  29. 10
  30. 7225
  31. 13
  32. 16
  33. 784
  34. 11025
  35. 4
  36. (x + 2)(x + 5)
  37. 181
  38. 7
  39. 48
  40. 25

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