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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 382 · 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.If a + b = 18 and ab = 72, what is (a − b)²? (a) 6 (b) 36 (c) 180 (d) 324
  2. 2.If a + b = 9 and ab = 20, what is (a − b)²? (a) 41 (b) 61 (c) 1 (d) 81
  3. 3.If a + b = 15 and ab = 56, what is a² + b²? (a) 337 (b) 225 (c) 113 (d) 169
  4. 4.What is the sum of the roots of x² − 10x + 21 = 0? (a) 9 (b) 10 (c) 11 (d) 21
  5. 5.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 63) (c) (x + 1)(x + 15) (d) (x + 7)(x + 9)
  6. 6.What is the product of the roots of x² − 7x + 10 = 0? (a) 7 (b) 11 (c) 9 (d) 10
  7. 7.What are the roots of x² − 12x + 27 = 0? (a) 2 and 10 (b) 1 and 27 (c) 1 and 11 (d) 3 and 9
  8. 8.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 10) (b) (x − 3)(x − 8) (c) (x + 1)(x + 24) (d) (x + 3)(x + 8)
  9. 9.What are the roots of x² − 15x + 56 = 0? (a) 7 and 8 (b) 1 and 56 (c) 1 and 14 (d) 2 and 28
  10. 10.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9996 (b) 9614 (c) 9604 (d) 10404
  11. 11.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 8099 (b) 7931 (c) 8281 (d) 7921
  12. 12.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 635 (b) 625 (c) 225 (d) 425
  13. 13.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 416 (b) 256 (c) 586 (d) 576
  14. 14.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 12) (b) (x + 4)(x + 9) (c) (x + 1)(x + 36) (d) (x − 4)(x − 9)
  15. 15.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 72 (c) 17 (d) 71
  16. 16.What is the sum of the roots of x² − 9x + 20 = 0? (a) 9 (b) 10 (c) 20 (d) 8
  17. 17.What is the sum of the roots of x² − 13x + 36 = 0? (a) 12 (b) 36 (c) 13 (d) 14
  18. 18.What are the roots of x² − 10x + 21 = 0? (a) 1 and 21 (b) 2 and 8 (c) 1 and 9 (d) 3 and 7
  19. 19.What is the sum of the roots of x² − 15x + 56 = 0? (a) 16 (b) 15 (c) 14 (d) 56
  20. 20.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 5929 (b) 6409 (c) 6899 (d) 6889
  21. 21.If a + b = 11 and ab = 30, what is a² + b²? (a) 121 (b) 91 (c) 61 (d) 181
  22. 22.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 7579 (b) 8649 (c) 7569 (d) 8091
  23. 23.What is the sum of the roots of x² − 10x + 16 = 0? (a) 9 (b) 11 (c) 10 (d) 16
  24. 24.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 625 (b) 225 (c) 235 (d) 375
  25. 25.If a + b = 15 and ab = 44, what is (a − b)²? (a) 137 (b) 225 (c) 49 (d) 7
  26. 26.If a + b = 15 and ab = 44, what is a² + b²? (a) 313 (b) 181 (c) 137 (d) 225
  27. 27.Which is the correct factorisation of x² + 10x + 16? (a) (x + 2)(x + 8) (b) (x + 1)(x + 9) (c) (x − 2)(x − 8) (d) (x + 1)(x + 16)
  28. 28.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 13) (b) (x + 1)(x + 45) (c) (x + 5)(x + 9) (d) (x − 5)(x − 9)
  29. 29.If a + b = 12 and ab = 20, what is a² + b²? (a) 124 (b) 104 (c) 144 (d) 184
  30. 30.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x − 6)(x − 7) (c) (x + 6)(x + 7) (d) (x + 1)(x + 42)
  31. 31.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 7066 (b) 7056 (c) 5776 (d) 6416
  32. 32.Which is the correct factorisation of x² + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 1)(x + 30) (c) (x + 1)(x + 10) (d) (x + 5)(x + 6)
  33. 33.If a + b = 17 and ab = 66, what is a² + b²? (a) 289 (b) 421 (c) 157 (d) 223
  34. 34.What are the roots of x² − 11x + 24 = 0? (a) 1 and 10 (b) 3 and 8 (c) 1 and 24 (d) 2 and 12
  35. 35.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 1024 (b) 1034 (c) 784 (d) 904
  36. 36.What are the roots of x² − 12x + 32 = 0? (a) 1 and 11 (b) 1 and 32 (c) 2 and 16 (d) 4 and 8
  37. 37.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2819 (b) 2809 (c) 2509 (d) 2209
  38. 38.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 72 (c) 17 (d) 16
  39. 39.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1946 (b) 1616 (c) 1936 (d) 1296
  40. 40.What is the sum of the roots of x² − 12x + 35 = 0? (a) 12 (b) 13 (c) 35 (d) 11

Answer key

  1. 36
  2. 1
  3. 113
  4. 10
  5. (x + 7)(x + 9)
  6. 10
  7. 3 and 9
  8. (x + 3)(x + 8)
  9. 7 and 8
  10. 9604
  11. 7921
  12. 625
  13. 576
  14. (x + 4)(x + 9)
  15. 72
  16. 9
  17. 13
  18. 3 and 7
  19. 15
  20. 6889
  21. 61
  22. 7569
  23. 10
  24. 225
  25. 49
  26. 137
  27. (x + 2)(x + 8)
  28. (x + 5)(x + 9)
  29. 104
  30. (x + 6)(x + 7)
  31. 7056
  32. (x + 5)(x + 6)
  33. 157
  34. 3 and 8
  35. 1024
  36. 4 and 8
  37. 2809
  38. 17
  39. 1936
  40. 12

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