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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 864 · 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 + 1)(x + 10) (b) (x + 5)(x + 6) (c) (x + 1)(x + 30) (d) (x − 5)(x − 6)
  2. 2.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)
  3. 3.What are the roots of x² − 11x + 30 = 0? (a) 5 and 6 (b) 1 and 30 (c) 1 and 10 (d) 2 and 15
  4. 4.What are the roots of x² − 7x + 10 = 0? (a) 1 and 10 (b) 2 and 5 (c) 1 and 6 (d) 3 and 4
  5. 5.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6571 (b) 6401 (c) 6561 (d) 6241
  6. 6.If a + b = 20 and ab = 99, what is a² + b²? (a) 400 (b) 202 (c) 301 (d) 598
  7. 7.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1764 (b) 1454 (c) 1444 (d) 1596
  8. 8.If a + b = 6 and ab = 8, what is a² + b²? (a) 28 (b) 52 (c) 20 (d) 36
  9. 9.If a + b = 18 and ab = 77, what is a² + b²? (a) 324 (b) 478 (c) 247 (d) 170
  10. 10.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 375 (b) 225 (c) 625 (d) 235
  11. 11.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 5929 (b) 6899 (c) 6889 (d) 6409
  12. 12.What is the sum of the roots of x² − 9x + 14 = 0? (a) 8 (b) 14 (c) 10 (d) 9
  13. 13.Which is the correct factorisation of x² + 8x + 12? (a) (x − 2)(x − 6) (b) (x + 2)(x + 6) (c) (x + 1)(x + 12) (d) (x + 1)(x + 7)
  14. 14.If a + b = 21 and ab = 108, what is a² + b²? (a) 441 (b) 225 (c) 333 (d) 657
  15. 15.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)
  16. 16.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3601 (b) 3731 (c) 3721 (d) 3481
  17. 17.What is the sum of the roots of x² − 7x + 10 = 0? (a) 8 (b) 7 (c) 6 (d) 10
  18. 18.If a + b = 18 and ab = 80, what is a² + b²? (a) 324 (b) 484 (c) 164 (d) 244
  19. 19.What are the roots of x² − 15x + 56 = 0? (a) 2 and 28 (b) 1 and 14 (c) 1 and 56 (d) 7 and 8
  20. 20.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 9) (b) (x − 3)(x − 7) (c) (x + 1)(x + 21) (d) (x + 3)(x + 7)
  21. 21.What is the product of the roots of x² − 10x + 16 = 0? (a) 16 (b) 15 (c) 17 (d) 10
  22. 22.What is the sum of the roots of x² − 17x + 72 = 0? (a) 17 (b) 16 (c) 18 (d) 72
  23. 23.If a + b = 15 and ab = 56, what is a² + b²? (a) 169 (b) 337 (c) 225 (d) 113
  24. 24.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 8) (b) (x + 4)(x + 5) (c) (x + 1)(x + 20) (d) (x − 4)(x − 5)
  25. 25.If a + b = 19 and ab = 90, what is a² + b²? (a) 541 (b) 271 (c) 181 (d) 361
  26. 26.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 12) (b) (x + 4)(x + 9) (c) (x − 4)(x − 9) (d) (x + 1)(x + 36)
  27. 27.If a + b = 12 and ab = 27, what is a² + b²? (a) 90 (b) 198 (c) 117 (d) 144
  28. 28.If a + b = 13 and ab = 12, what is a² + b²? (a) 145 (b) 193 (c) 169 (d) 157
  29. 29.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 729 (b) 909 (c) 1099 (d) 1089
  30. 30.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 971 (b) 901 (c) 961 (d) 841
  31. 31.Which is the correct factorisation of x² + 11x + 18? (a) (x − 2)(x − 9) (b) (x + 1)(x + 18) (c) (x + 1)(x + 10) (d) (x + 2)(x + 9)
  32. 32.If a + b = 8 and ab = 15, what is a² + b²? (a) 49 (b) 34 (c) 94 (d) 64
  33. 33.If a + b = 17 and ab = 70, what is (a − b)²? (a) 9 (b) 149 (c) 3 (d) 289
  34. 34.If a + b = 12 and ab = 11, what is a² + b²? (a) 144 (b) 166 (c) 122 (d) 133
  35. 35.What are the roots of x² − 16x + 63 = 0? (a) 3 and 21 (b) 7 and 9 (c) 1 and 15 (d) 1 and 63
  36. 36.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x + 1)(x + 12) (c) (x + 6)(x + 7) (d) (x − 6)(x − 7)
  37. 37.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 841 (b) 899 (c) 851 (d) 961
  38. 38.If a + b = 10 and ab = 21, what is (a − b)²? (a) 100 (b) 16 (c) 4 (d) 58
  39. 39.What are the roots of x² − 10x + 21 = 0? (a) 1 and 9 (b) 2 and 8 (c) 3 and 7 (d) 1 and 21
  40. 40.If a + b = 14 and ab = 40, what is (a − b)²? (a) 6 (b) 116 (c) 36 (d) 196

Answer key

  1. (x + 5)(x + 6)
  2. (x + 3)(x + 4)
  3. 5 and 6
  4. 2 and 5
  5. 6561
  6. 202
  7. 1444
  8. 20
  9. 170
  10. 225
  11. 6889
  12. 9
  13. (x + 2)(x + 6)
  14. 225
  15. (x + 5)(x + 8)
  16. 3721
  17. 7
  18. 164
  19. 7 and 8
  20. (x + 3)(x + 7)
  21. 16
  22. 17
  23. 113
  24. (x + 4)(x + 5)
  25. 181
  26. (x + 4)(x + 9)
  27. 90
  28. 145
  29. 1089
  30. 961
  31. (x + 2)(x + 9)
  32. 34
  33. 9
  34. 122
  35. 7 and 9
  36. (x + 6)(x + 7)
  37. 841
  38. 16
  39. 3 and 7
  40. 36

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