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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 300 · 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.If a + b = 18 and ab = 77, what is a² + b²? (a) 478 (b) 247 (c) 170 (d) 324
  3. 3.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 9025 (b) 8125 (c) 9035 (d) 7225
  4. 4.Which is the correct factorisation of x² + 9x + 14? (a) (x − 2)(x − 7) (b) (x + 1)(x + 8) (c) (x + 2)(x + 7) (d) (x + 1)(x + 14)
  5. 5.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 7)(x + 8) (c) (x + 1)(x + 14) (d) (x − 7)(x − 8)
  6. 6.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 6)(x + 8) (c) (x + 1)(x + 48) (d) (x + 1)(x + 13)
  7. 7.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1296 (b) 1584 (c) 1306 (d) 1936
  8. 8.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4899 (b) 5041 (c) 4771 (d) 4761
  9. 9.If a + b = 19 and ab = 88, what is a² + b²? (a) 361 (b) 537 (c) 273 (d) 185
  10. 10.What is the sum of the roots of x² − 15x + 54 = 0? (a) 16 (b) 54 (c) 14 (d) 15
  11. 11.If a + b = 8 and ab = 15, what is a² + b²? (a) 49 (b) 94 (c) 34 (d) 64
  12. 12.Which is the correct factorisation of x² + 11x + 18? (a) (x + 2)(x + 9) (b) (x + 1)(x + 10) (c) (x − 2)(x − 9) (d) (x + 1)(x + 18)
  13. 13.What is the sum of the roots of x² − 17x + 72 = 0? (a) 17 (b) 16 (c) 72 (d) 18
  14. 14.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 12) (b) (x + 1)(x + 40) (c) (x − 5)(x − 8) (d) (x + 5)(x + 8)
  15. 15.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 8)(x + 9) (c) (x + 1)(x + 72) (d) (x − 8)(x − 9)
  16. 16.What are the roots of x² − 14x + 48 = 0? (a) 2 and 24 (b) 1 and 13 (c) 6 and 8 (d) 1 and 48
  17. 17.If a + b = 16 and ab = 48, what is a² + b²? (a) 256 (b) 208 (c) 160 (d) 352
  18. 18.Which is the correct factorisation of x² + 12x + 32? (a) (x − 4)(x − 8) (b) (x + 1)(x + 32) (c) (x + 4)(x + 8) (d) (x + 1)(x + 11)
  19. 19.If a + b = 13 and ab = 30, what is a² + b²? (a) 169 (b) 229 (c) 139 (d) 109
  20. 20.What are the roots of x² − 15x + 56 = 0? (a) 1 and 14 (b) 2 and 28 (c) 7 and 8 (d) 1 and 56
  21. 21.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10201 (b) 10211 (c) 9801 (d) 10001
  22. 22.What is the sum of the roots of x² − 11x + 24 = 0? (a) 10 (b) 11 (c) 24 (d) 12
  23. 23.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10404 (b) 10004 (c) 10414 (d) 9604
  24. 24.If a + b = 10 and ab = 24, what is (a − b)²? (a) 52 (b) 4 (c) 100 (d) 2
  25. 25.What is the sum of the roots of x² − 16x + 63 = 0? (a) 16 (b) 63 (c) 17 (d) 15
  26. 26.Which is the correct factorisation of x² + 13x + 42? (a) (x + 6)(x + 7) (b) (x + 1)(x + 12) (c) (x + 1)(x + 42) (d) (x − 6)(x − 7)
  27. 27.What is the sum of the roots of x² − 8x + 12 = 0? (a) 8 (b) 7 (c) 12 (d) 9
  28. 28.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)
  29. 29.What is the product of the roots of x² − 8x + 12 = 0? (a) 12 (b) 13 (c) 8 (d) 11
  30. 30.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 3584 (b) 3136 (c) 3146 (d) 4096
  31. 31.Which is the correct factorisation of x² + 6x + 8? (a) (x + 2)(x + 4) (b) (x + 1)(x + 5) (c) (x + 1)(x + 8) (d) (x − 2)(x − 4)
  32. 32.If a + b = 8 and ab = 7, what is a² + b²? (a) 50 (b) 57 (c) 78 (d) 64
  33. 33.If a + b = 4 and ab = 3, what is a² + b²? (a) 13 (b) 16 (c) 22 (d) 10
  34. 34.What are the roots of x² − 8x + 12 = 0? (a) 1 and 12 (b) 3 and 4 (c) 1 and 7 (d) 2 and 6
  35. 35.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 925 (b) 1225 (c) 625 (d) 1235
  36. 36.Which is the correct factorisation of x² + 12x + 35? (a) (x − 5)(x − 7) (b) (x + 5)(x + 7) (c) (x + 1)(x + 11) (d) (x + 1)(x + 35)
  37. 37.What is the sum of the roots of x² − 12x + 27 = 0? (a) 13 (b) 12 (c) 27 (d) 11
  38. 38.What is the product of the roots of x² − 15x + 56 = 0? (a) 15 (b) 55 (c) 56 (d) 57
  39. 39.What is the product of the roots of x² − 16x + 63 = 0? (a) 64 (b) 62 (c) 63 (d) 16
  40. 40.If a + b = 10 and ab = 24, what is a² + b²? (a) 148 (b) 52 (c) 100 (d) 76

Answer key

  1. 8
  2. 170
  3. 9025
  4. (x + 2)(x + 7)
  5. (x + 7)(x + 8)
  6. (x + 6)(x + 8)
  7. 1296
  8. 4761
  9. 185
  10. 15
  11. 34
  12. (x + 2)(x + 9)
  13. 17
  14. (x + 5)(x + 8)
  15. (x + 8)(x + 9)
  16. 6 and 8
  17. 160
  18. (x + 4)(x + 8)
  19. 109
  20. 7 and 8
  21. 10201
  22. 11
  23. 10404
  24. 4
  25. 16
  26. (x + 6)(x + 7)
  27. 8
  28. (x + 3)(x + 9)
  29. 12
  30. 3136
  31. (x + 2)(x + 4)
  32. 50
  33. 10
  34. 2 and 6
  35. 1225
  36. (x + 5)(x + 7)
  37. 12
  38. 56
  39. 63
  40. 52

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