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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 497 · 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.Which is the correct factorisation of x² + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 3)(x + 4) (c) (x + 1)(x + 12) (d) (x + 1)(x + 6)
  2. 2.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 28) (b) (x + 4)(x + 7) (c) (x + 1)(x + 10) (d) (x − 4)(x − 7)
  3. 3.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 3625 (b) 3025 (c) 4235 (d) 4225
  4. 4.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10004 (b) 9604 (c) 10404 (d) 10414
  5. 5.If a + b = 12 and ab = 11, what is (a − b)²? (a) 122 (b) 144 (c) 10 (d) 100
  6. 6.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 1)(x + 12) (c) (x + 6)(x + 7) (d) (x + 1)(x + 42)
  7. 7.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 4225 (b) 5635 (c) 5625 (d) 4925
  8. 8.What is the product of the roots of x² − 9x + 20 = 0? (a) 19 (b) 9 (c) 21 (d) 20
  9. 9.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 971 (b) 841 (c) 961 (d) 901
  10. 10.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1609 (b) 1859 (c) 1849 (d) 1369
  11. 11.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 1)(x + 16) (c) (x − 8)(x − 9) (d) (x + 8)(x + 9)
  12. 12.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 10) (b) (x + 1)(x + 6) (c) (x − 2)(x − 5) (d) (x + 2)(x + 5)
  13. 13.If a + b = 18 and ab = 77, what is (a − b)²? (a) 324 (b) 16 (c) 4 (d) 170
  14. 14.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x − 7)(x − 8) (c) (x + 1)(x + 56) (d) (x + 7)(x + 8)
  15. 15.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6396 (b) 6724 (c) 6084 (d) 6094
  16. 16.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 7) (b) (x − 3)(x − 5) (c) (x + 1)(x + 15) (d) (x + 3)(x + 5)
  17. 17.What are the roots of x² − 8x + 12 = 0? (a) 1 and 12 (b) 1 and 7 (c) 3 and 4 (d) 2 and 6
  18. 18.What are the roots of x² − 11x + 18 = 0? (a) 1 and 18 (b) 1 and 10 (c) 3 and 6 (d) 2 and 9
  19. 19.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)
  20. 20.What is the sum of the roots of x² − 10x + 16 = 0? (a) 10 (b) 11 (c) 9 (d) 16
  21. 21.What is the product of the roots of x² − 10x + 24 = 0? (a) 25 (b) 10 (c) 24 (d) 23
  22. 22.What is the product of the roots of x² − 12x + 32 = 0? (a) 31 (b) 32 (c) 33 (d) 12
  23. 23.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 45) (b) (x + 1)(x + 13) (c) (x − 5)(x − 9) (d) (x + 5)(x + 9)
  24. 24.If a + b = 12 and ab = 27, what is a² + b²? (a) 144 (b) 198 (c) 90 (d) 117
  25. 25.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 1156 (b) 676 (c) 1166 (d) 916
  26. 26.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9035 (b) 9975 (c) 9025 (d) 11025
  27. 27.If a + b = 10 and ab = 9, what is a² + b²? (a) 118 (b) 100 (c) 91 (d) 82
  28. 28.What is the product of the roots of x² − 13x + 42 = 0? (a) 13 (b) 41 (c) 43 (d) 42
  29. 29.What is the sum of the roots of x² − 7x + 10 = 0? (a) 6 (b) 8 (c) 10 (d) 7
  30. 30.What is the sum of the roots of x² − 5x + 6 = 0? (a) 6 (b) 5 (c) 7 (d) 4
  31. 31.If a + b = 15 and ab = 56, what is (a − b)²? (a) 169 (b) 113 (c) 1 (d) 225
  32. 32.What is the sum of the roots of x² − 14x + 45 = 0? (a) 15 (b) 14 (c) 13 (d) 45
  33. 33.If a + b = 18 and ab = 77, what is a² + b²? (a) 478 (b) 247 (c) 324 (d) 170
  34. 34.If a + b = 16 and ab = 63, what is a² + b²? (a) 256 (b) 382 (c) 193 (d) 130
  35. 35.If a + b = 9 and ab = 8, what is a² + b²? (a) 81 (b) 65 (c) 97 (d) 73
  36. 36.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1521 (b) 1599 (c) 1681 (d) 1531
  37. 37.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2509 (b) 2809 (c) 2819 (d) 2209
  38. 38.If a + b = 14 and ab = 48, what is a² + b²? (a) 196 (b) 292 (c) 100 (d) 148
  39. 39.If a + b = 10 and ab = 16, what is a² + b²? (a) 100 (b) 132 (c) 84 (d) 68
  40. 40.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 14 (c) 16 (d) 15

Answer key

  1. (x + 3)(x + 4)
  2. (x + 4)(x + 7)
  3. 4225
  4. 10404
  5. 100
  6. (x + 6)(x + 7)
  7. 5625
  8. 20
  9. 961
  10. 1849
  11. (x + 8)(x + 9)
  12. (x + 2)(x + 5)
  13. 16
  14. (x + 7)(x + 8)
  15. 6084
  16. (x + 3)(x + 5)
  17. 2 and 6
  18. 2 and 9
  19. (x + 4)(x + 8)
  20. 10
  21. 24
  22. 32
  23. (x + 5)(x + 9)
  24. 90
  25. 1156
  26. 9025
  27. 82
  28. 42
  29. 7
  30. 5
  31. 1
  32. 14
  33. 170
  34. 130
  35. 65
  36. 1521
  37. 2809
  38. 100
  39. 68
  40. 15

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