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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 704 · 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 + 15x + 56? (a) (x + 1)(x + 14) (b) (x − 7)(x − 8) (c) (x + 1)(x + 56) (d) (x + 7)(x + 8)
  2. 2.If a + b = 18 and ab = 72, what is (a − b)²? (a) 324 (b) 6 (c) 36 (d) 180
  3. 3.What is the sum of the roots of x² − 7x + 12 = 0? (a) 8 (b) 12 (c) 7 (d) 6
  4. 4.If a + b = 5 and ab = 6, what is a² + b²? (a) 19 (b) 25 (c) 37 (d) 13
  5. 5.If a + b = 10 and ab = 24, what is a² + b²? (a) 148 (b) 100 (c) 52 (d) 76
  6. 6.What are the roots of x² − 16x + 63 = 0? (a) 7 and 9 (b) 1 and 63 (c) 3 and 21 (d) 1 and 15
  7. 7.What is the sum of the roots of x² − 15x + 56 = 0? (a) 15 (b) 56 (c) 14 (d) 16
  8. 8.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 375 (b) 225 (c) 625 (d) 235
  9. 9.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6401 (b) 6571 (c) 6241 (d) 6561
  10. 10.If a + b = 21 and ab = 108, what is a² + b²? (a) 441 (b) 333 (c) 657 (d) 225
  11. 11.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 1)(x + 13) (c) (x + 6)(x + 8) (d) (x + 1)(x + 48)
  12. 12.Which is the correct factorisation of x² + 5x + 6? (a) (x + 2)(x + 3) (b) (x + 1)(x + 4) (c) (x − 2)(x − 3) (d) (x + 1)(x + 6)
  13. 13.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 40) (b) (x + 1)(x + 12) (c) (x + 5)(x + 8) (d) (x − 5)(x − 8)
  14. 14.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)
  15. 15.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 16 (c) 18 (d) 17
  16. 16.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6399 (b) 6561 (c) 6251 (d) 6241
  17. 17.If a + b = 19 and ab = 88, what is (a − b)²? (a) 3 (b) 9 (c) 185 (d) 361
  18. 18.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 3625 (b) 4225 (c) 3025 (d) 4235
  19. 19.If a + b = 9 and ab = 18, what is a² + b²? (a) 117 (b) 63 (c) 45 (d) 81
  20. 20.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)
  21. 21.What are the roots of x² − 15x + 56 = 0? (a) 2 and 28 (b) 7 and 8 (c) 1 and 14 (d) 1 and 56
  22. 22.If a + b = 14 and ab = 45, what is (a − b)²? (a) 106 (b) 4 (c) 196 (d) 16
  23. 23.If a + b = 14 and ab = 40, what is a² + b²? (a) 196 (b) 276 (c) 116 (d) 156
  24. 24.If a + b = 11 and ab = 30, what is (a − b)²? (a) 91 (b) 61 (c) 121 (d) 1
  25. 25.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 3136 (b) 4106 (c) 4096 (d) 3616
  26. 26.If a + b = 12 and ab = 20, what is (a − b)²? (a) 64 (b) 104 (c) 144 (d) 8
  27. 27.If a + b = 18 and ab = 77, what is a² + b²? (a) 247 (b) 324 (c) 170 (d) 478
  28. 28.Which is the correct factorisation of x² + 15x + 54? (a) (x − 6)(x − 9) (b) (x + 6)(x + 9) (c) (x + 1)(x + 54) (d) (x + 1)(x + 14)
  29. 29.If a + b = 6 and ab = 5, what is (a − b)²? (a) 4 (b) 26 (c) 16 (d) 36
  30. 30.Which is the correct factorisation of x² + 9x + 20? (a) (x + 4)(x + 5) (b) (x + 1)(x + 8) (c) (x − 4)(x − 5) (d) (x + 1)(x + 20)
  31. 31.What are the roots of x² − 13x + 42 = 0? (a) 1 and 12 (b) 1 and 42 (c) 2 and 21 (d) 6 and 7
  32. 32.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 324 (b) 404 (c) 484 (d) 494
  33. 33.What is the product of the roots of x² − 9x + 20 = 0? (a) 21 (b) 19 (c) 9 (d) 20
  34. 34.If a + b = 18 and ab = 80, what is (a − b)²? (a) 2 (b) 324 (c) 4 (d) 164
  35. 35.What is the product of the roots of x² − 12x + 35 = 0? (a) 34 (b) 12 (c) 35 (d) 36
  36. 36.If a + b = 12 and ab = 35, what is (a − b)²? (a) 2 (b) 74 (c) 4 (d) 144
  37. 37.What is the product of the roots of x² − 9x + 14 = 0? (a) 13 (b) 9 (c) 14 (d) 15
  38. 38.What is the product of the roots of x² − 7x + 12 = 0? (a) 12 (b) 13 (c) 11 (d) 7
  39. 39.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 9226 (b) 9984 (c) 10816 (d) 9216
  40. 40.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1599 (b) 1681 (c) 1531 (d) 1521

Answer key

  1. (x + 7)(x + 8)
  2. 36
  3. 7
  4. 13
  5. 52
  6. 7 and 9
  7. 15
  8. 225
  9. 6561
  10. 225
  11. (x + 6)(x + 8)
  12. (x + 2)(x + 3)
  13. (x + 5)(x + 8)
  14. (x + 8)(x + 9)
  15. 17
  16. 6241
  17. 9
  18. 4225
  19. 45
  20. (x + 4)(x + 7)
  21. 7 and 8
  22. 16
  23. 116
  24. 1
  25. 4096
  26. 64
  27. 170
  28. (x + 6)(x + 9)
  29. 16
  30. (x + 4)(x + 5)
  31. 6 and 7
  32. 484
  33. 20
  34. 4
  35. 35
  36. 4
  37. 14
  38. 12
  39. 9216
  40. 1521

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