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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 688 · 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 + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 5)(x + 6) (c) (x + 1)(x + 30) (d) (x + 1)(x + 10)
  2. 2.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 72 (c) 73 (d) 71
  3. 3.Which is the correct factorisation of x² + 8x + 12? (a) (x − 2)(x − 6) (b) (x + 1)(x + 12) (c) (x + 1)(x + 7) (d) (x + 2)(x + 6)
  4. 4.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 35) (b) (x − 5)(x − 7) (c) (x + 1)(x + 11) (d) (x + 5)(x + 7)
  5. 5.If a + b = 7 and ab = 6, what is (a − b)²? (a) 37 (b) 5 (c) 25 (d) 49
  6. 6.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 9025 (b) 8075 (c) 7225 (d) 7235
  7. 7.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x + 1)(x + 63) (c) (x − 7)(x − 9) (d) (x + 1)(x + 15)
  8. 8.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 10) (b) (x − 4)(x − 7) (c) (x + 1)(x + 28) (d) (x + 4)(x + 7)
  9. 9.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)
  10. 10.If a + b = 19 and ab = 84, what is (a − b)²? (a) 361 (b) 5 (c) 193 (d) 25
  11. 11.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 3136 (b) 4096 (c) 4106 (d) 3616
  12. 12.If a + b = 12 and ab = 20, what is a² + b²? (a) 184 (b) 124 (c) 144 (d) 104
  13. 13.What is the sum of the roots of x² − 8x + 12 = 0? (a) 8 (b) 12 (c) 9 (d) 7
  14. 14.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4761 (b) 5041 (c) 4899 (d) 4771
  15. 15.What is the product of the roots of x² − 10x + 16 = 0? (a) 15 (b) 17 (c) 16 (d) 10
  16. 16.What is the product of the roots of x² − 10x + 24 = 0? (a) 24 (b) 10 (c) 25 (d) 23
  17. 17.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2704 (b) 2314 (c) 2304 (d) 2496
  18. 18.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 2025 (b) 2475 (c) 2035 (d) 3025
  19. 19.Which is the correct factorisation of x² + 14x + 45? (a) (x + 5)(x + 9) (b) (x + 1)(x + 45) (c) (x + 1)(x + 13) (d) (x − 5)(x − 9)
  20. 20.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 529 (b) 391 (c) 299 (d) 289
  21. 21.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 9) (b) (x + 1)(x + 21) (c) (x + 3)(x + 7) (d) (x − 3)(x − 7)
  22. 22.What is the sum of the roots of x² − 15x + 56 = 0? (a) 15 (b) 56 (c) 16 (d) 14
  23. 23.If a + b = 17 and ab = 70, what is (a − b)²? (a) 9 (b) 289 (c) 149 (d) 3
  24. 24.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 14) (b) (x + 1)(x + 8) (c) (x + 2)(x + 7) (d) (x − 2)(x − 7)
  25. 25.What is the product of the roots of x² − 13x + 40 = 0? (a) 39 (b) 40 (c) 41 (d) 13
  26. 26.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 961 (b) 901 (c) 841 (d) 971
  27. 27.If a + b = 10 and ab = 9, what is a² + b²? (a) 91 (b) 100 (c) 118 (d) 82
  28. 28.If a + b = 5 and ab = 4, what is (a − b)²? (a) 9 (b) 25 (c) 3 (d) 17
  29. 29.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 4) (b) (x + 2)(x + 3) (c) (x + 1)(x + 6) (d) (x − 2)(x − 3)
  30. 30.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 6889 (b) 5939 (c) 5929 (d) 6391
  31. 31.If a + b = 15 and ab = 36, what is a² + b²? (a) 225 (b) 297 (c) 189 (d) 153
  32. 32.What is the sum of the roots of x² − 6x + 8 = 0? (a) 8 (b) 5 (c) 6 (d) 7
  33. 33.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 27) (b) (x − 3)(x − 9) (c) (x + 1)(x + 11) (d) (x + 3)(x + 9)
  34. 34.If a + b = 9 and ab = 18, what is a² + b²? (a) 81 (b) 45 (c) 63 (d) 117
  35. 35.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 8125 (b) 9035 (c) 7225 (d) 9025
  36. 36.What is the product of the roots of x² − 8x + 12 = 0? (a) 13 (b) 8 (c) 11 (d) 12
  37. 37.What is the product of the roots of x² − 6x + 8 = 0? (a) 9 (b) 7 (c) 8 (d) 6
  38. 38.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 72 (c) 17 (d) 18
  39. 39.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 5635 (b) 7225 (c) 5625 (d) 6375
  40. 40.If a + b = 8 and ab = 15, what is a² + b²? (a) 34 (b) 49 (c) 94 (d) 64

Answer key

  1. (x + 5)(x + 6)
  2. 72
  3. (x + 2)(x + 6)
  4. (x + 5)(x + 7)
  5. 25
  6. 7225
  7. (x + 7)(x + 9)
  8. (x + 4)(x + 7)
  9. (x + 7)(x + 8)
  10. 25
  11. 4096
  12. 104
  13. 8
  14. 4761
  15. 16
  16. 24
  17. 2304
  18. 2025
  19. (x + 5)(x + 9)
  20. 289
  21. (x + 3)(x + 7)
  22. 15
  23. 9
  24. (x + 2)(x + 7)
  25. 40
  26. 961
  27. 82
  28. 9
  29. (x + 2)(x + 3)
  30. 5929
  31. 153
  32. 6
  33. (x + 3)(x + 9)
  34. 45
  35. 9025
  36. 12
  37. 8
  38. 17
  39. 5625
  40. 34

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