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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 590 · 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 + 7x + 10? (a) (x + 2)(x + 5) (b) (x + 1)(x + 6) (c) (x + 1)(x + 10) (d) (x − 2)(x − 5)
  2. 2.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 635 (b) 425 (c) 625 (d) 225
  3. 3.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1601 (b) 1691 (c) 1521 (d) 1681
  4. 4.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 484 (b) 404 (c) 494 (d) 324
  5. 5.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 9) (b) (x − 4)(x − 6) (c) (x + 1)(x + 24) (d) (x + 4)(x + 6)
  6. 6.What is the sum of the roots of x² − 10x + 21 = 0? (a) 11 (b) 10 (c) 21 (d) 9
  7. 7.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 7)(x + 9) (c) (x + 1)(x + 63) (d) (x − 7)(x − 9)
  8. 8.If a + b = 19 and ab = 90, what is (a − b)²? (a) 181 (b) 1 (c) 361 (d) 271
  9. 9.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6241 (b) 6399 (c) 6251 (d) 6561
  10. 10.What is the sum of the roots of x² − 9x + 14 = 0? (a) 8 (b) 9 (c) 14 (d) 10
  11. 11.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3249 (b) 3969 (c) 3979 (d) 3609
  12. 12.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9614 (b) 9604 (c) 10404 (d) 9996
  13. 13.What is the sum of the roots of x² − 7x + 12 = 0? (a) 12 (b) 8 (c) 7 (d) 6
  14. 14.If a + b = 20 and ab = 96, what is a² + b²? (a) 400 (b) 592 (c) 304 (d) 208
  15. 15.If a + b = 13 and ab = 40, what is a² + b²? (a) 129 (b) 89 (c) 249 (d) 169
  16. 16.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 3616 (b) 4106 (c) 3136 (d) 4096
  17. 17.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 3035 (b) 3025 (c) 4225 (d) 3575
  18. 18.What is the sum of the roots of x² − 16x + 63 = 0? (a) 15 (b) 17 (c) 63 (d) 16
  19. 19.What is the sum of the roots of x² − 8x + 15 = 0? (a) 7 (b) 8 (c) 9 (d) 15
  20. 20.What are the roots of x² − 14x + 45 = 0? (a) 1 and 13 (b) 3 and 15 (c) 5 and 9 (d) 1 and 45
  21. 21.What are the roots of x² − 9x + 20 = 0? (a) 1 and 20 (b) 1 and 8 (c) 4 and 5 (d) 2 and 10
  22. 22.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10004 (b) 10414 (c) 9604 (d) 10404
  23. 23.If a + b = 15 and ab = 44, what is (a − b)²? (a) 7 (b) 225 (c) 137 (d) 49
  24. 24.What are the roots of x² − 17x + 72 = 0? (a) 8 and 9 (b) 2 and 36 (c) 1 and 72 (d) 1 and 16
  25. 25.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 1156 (b) 1166 (c) 916 (d) 676
  26. 26.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 875 (b) 635 (c) 625 (d) 1225
  27. 27.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 4875 (b) 4225 (c) 4235 (d) 5625
  28. 28.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 5184 (b) 4624 (c) 4634 (d) 4896
  29. 29.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 40) (b) (x + 5)(x + 8) (c) (x + 1)(x + 12) (d) (x − 5)(x − 8)
  30. 30.Which is the correct factorisation of x² + 14x + 48? (a) (x + 6)(x + 8) (b) (x + 1)(x + 48) (c) (x − 6)(x − 8) (d) (x + 1)(x + 13)
  31. 31.What is the product of the roots of x² − 14x + 45 = 0? (a) 46 (b) 14 (c) 44 (d) 45
  32. 32.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2509 (b) 2819 (c) 2809 (d) 2209
  33. 33.If a + b = 3 and ab = 2, what is (a − b)²? (a) 7 (b) 5 (c) 9 (d) 1
  34. 34.Which is the correct factorisation of x² + 9x + 18? (a) (x − 3)(x − 6) (b) (x + 1)(x + 8) (c) (x + 1)(x + 18) (d) (x + 3)(x + 6)
  35. 35.What are the roots of x² − 11x + 24 = 0? (a) 1 and 10 (b) 2 and 12 (c) 1 and 24 (d) 3 and 8
  36. 36.What are the roots of x² − 11x + 30 = 0? (a) 1 and 10 (b) 2 and 15 (c) 1 and 30 (d) 5 and 6
  37. 37.If a + b = 7 and ab = 10, what is (a − b)²? (a) 49 (b) 9 (c) 29 (d) 3
  38. 38.What are the roots of x² − 8x + 12 = 0? (a) 1 and 7 (b) 1 and 12 (c) 3 and 4 (d) 2 and 6
  39. 39.What are the roots of x² − 7x + 12 = 0? (a) 1 and 12 (b) 3 and 4 (c) 2 and 6 (d) 1 and 6
  40. 40.If a + b = 15 and ab = 36, what is (a − b)²? (a) 153 (b) 225 (c) 81 (d) 9

Answer key

  1. (x + 2)(x + 5)
  2. 625
  3. 1681
  4. 484
  5. (x + 4)(x + 6)
  6. 10
  7. (x + 7)(x + 9)
  8. 1
  9. 6241
  10. 9
  11. 3969
  12. 9604
  13. 7
  14. 208
  15. 89
  16. 4096
  17. 3025
  18. 16
  19. 8
  20. 5 and 9
  21. 4 and 5
  22. 10404
  23. 49
  24. 8 and 9
  25. 1156
  26. 625
  27. 4225
  28. 4624
  29. (x + 5)(x + 8)
  30. (x + 6)(x + 8)
  31. 45
  32. 2809
  33. 1
  34. (x + 3)(x + 6)
  35. 3 and 8
  36. 5 and 6
  37. 9
  38. 2 and 6
  39. 3 and 4
  40. 81

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