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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 966 · 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 sum of the roots of x² − 11x + 18 = 0? (a) 10 (b) 12 (c) 18 (d) 11
  2. 2.What is the product of the roots of x² − 7x + 10 = 0? (a) 9 (b) 10 (c) 11 (d) 7
  3. 3.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x − 7)(x − 9) (c) (x + 1)(x + 15) (d) (x + 7)(x + 9)
  4. 4.If a + b = 3 and ab = 2, what is a² + b²? (a) 5 (b) 9 (c) 7 (d) 13
  5. 5.What is the product of the roots of x² − 10x + 24 = 0? (a) 24 (b) 25 (c) 23 (d) 10
  6. 6.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 18 (c) 72 (d) 17
  7. 7.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 441 (b) 451 (c) 401 (d) 361
  8. 8.What are the roots of x² − 16x + 63 = 0? (a) 1 and 15 (b) 7 and 9 (c) 1 and 63 (d) 3 and 21
  9. 9.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 9216 (b) 9226 (c) 9984 (d) 10816
  10. 10.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 12) (b) (x + 1)(x + 7) (c) (x + 2)(x + 6) (d) (x − 2)(x − 6)
  11. 11.If a + b = 11 and ab = 10, what is a² + b²? (a) 111 (b) 141 (c) 101 (d) 121
  12. 12.What is the sum of the roots of x² − 5x + 6 = 0? (a) 7 (b) 6 (c) 5 (d) 4
  13. 13.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)
  14. 14.What are the roots of x² − 14x + 48 = 0? (a) 1 and 48 (b) 1 and 13 (c) 6 and 8 (d) 2 and 24
  15. 15.If a + b = 10 and ab = 24, what is (a − b)²? (a) 100 (b) 2 (c) 4 (d) 52
  16. 16.If a + b = 15 and ab = 56, what is a² + b²? (a) 337 (b) 169 (c) 113 (d) 225
  17. 17.If a + b = 8 and ab = 12, what is a² + b²? (a) 88 (b) 64 (c) 52 (d) 40
  18. 18.If a + b = 10 and ab = 24, what is a² + b²? (a) 76 (b) 148 (c) 100 (d) 52
  19. 19.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x − 7)(x − 8) (c) (x + 7)(x + 8) (d) (x + 1)(x + 14)
  20. 20.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 5486 (b) 5476 (c) 4356 (d) 4916
  21. 21.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3969 (b) 3979 (c) 3249 (d) 3609
  22. 22.What is the product of the roots of x² − 12x + 35 = 0? (a) 35 (b) 36 (c) 12 (d) 34
  23. 23.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 896 (b) 1024 (c) 794 (d) 784
  24. 24.If a + b = 13 and ab = 30, what is (a − b)²? (a) 7 (b) 109 (c) 49 (d) 169
  25. 25.Which is the correct factorisation of x² + 11x + 18? (a) (x − 2)(x − 9) (b) (x + 2)(x + 9) (c) (x + 1)(x + 18) (d) (x + 1)(x + 10)
  26. 26.If a + b = 18 and ab = 80, what is a² + b²? (a) 324 (b) 484 (c) 164 (d) 244
  27. 27.What are the roots of x² − 5x + 6 = 0? (a) 1 and 4 (b) 2 and 3 (c) 2 and 4 (d) 1 and 6
  28. 28.What are the roots of x² − 9x + 14 = 0? (a) 1 and 14 (b) 3 and 6 (c) 1 and 8 (d) 2 and 7
  29. 29.Which is the correct factorisation of x² + 10x + 21? (a) (x − 3)(x − 7) (b) (x + 1)(x + 21) (c) (x + 1)(x + 9) (d) (x + 3)(x + 7)
  30. 30.What is the sum of the roots of x² − 14x + 45 = 0? (a) 45 (b) 13 (c) 15 (d) 14
  31. 31.What is the product of the roots of x² − 14x + 45 = 0? (a) 14 (b) 44 (c) 45 (d) 46
  32. 32.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x − 6)(x − 7) (c) (x + 6)(x + 7) (d) (x + 1)(x + 12)
  33. 33.What is the sum of the roots of x² − 10x + 21 = 0? (a) 9 (b) 10 (c) 21 (d) 11
  34. 34.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10009 (b) 9409 (c) 10619 (d) 10609
  35. 35.What is the sum of the roots of x² − 13x + 42 = 0? (a) 12 (b) 13 (c) 14 (d) 42
  36. 36.If a + b = 5 and ab = 6, what is (a − b)²? (a) 25 (b) 1 (c) 13 (d) 19
  37. 37.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 1)(x + 72) (c) (x + 8)(x + 9) (d) (x − 8)(x − 9)
  38. 38.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2819 (b) 2509 (c) 2809 (d) 2209
  39. 39.If a + b = 14 and ab = 24, what is a² + b²? (a) 172 (b) 148 (c) 196 (d) 244
  40. 40.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x − 5)(x − 6) (c) (x + 5)(x + 6) (d) (x + 1)(x + 30)

Answer key

  1. 11
  2. 10
  3. (x + 7)(x + 9)
  4. 5
  5. 24
  6. 17
  7. 441
  8. 7 and 9
  9. 9216
  10. (x + 2)(x + 6)
  11. 101
  12. 5
  13. (x + 2)(x + 3)
  14. 6 and 8
  15. 4
  16. 113
  17. 40
  18. 52
  19. (x + 7)(x + 8)
  20. 5476
  21. 3969
  22. 35
  23. 784
  24. 49
  25. (x + 2)(x + 9)
  26. 164
  27. 2 and 3
  28. 2 and 7
  29. (x + 3)(x + 7)
  30. 14
  31. 45
  32. (x + 6)(x + 7)
  33. 10
  34. 10609
  35. 13
  36. 1
  37. (x + 8)(x + 9)
  38. 2809
  39. 148
  40. (x + 5)(x + 6)

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