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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 62 · 40 questions
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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 = 8 and ab = 15, what is a² + b²? (a) 34 (b) 49 (c) 94 (d) 64
  2. 2.If a + b = 6 and ab = 5, what is (a − b)²? (a) 36 (b) 16 (c) 26 (d) 4
  3. 3.Which is the correct factorisation of x² + 13x + 40? (a) (x − 5)(x − 8) (b) (x + 5)(x + 8) (c) (x + 1)(x + 40) (d) (x + 1)(x + 12)
  4. 4.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6734 (b) 6404 (c) 6084 (d) 6724
  5. 5.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 5184 (b) 5194 (c) 4624 (d) 4904
  6. 6.If a + b = 7 and ab = 6, what is (a − b)²? (a) 25 (b) 37 (c) 49 (d) 5
  7. 7.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 5) (b) (x + 1)(x + 8) (c) (x + 2)(x + 4) (d) (x − 2)(x − 4)
  8. 8.What is the product of the roots of x² − 11x + 30 = 0? (a) 31 (b) 29 (c) 30 (d) 11
  9. 9.If a + b = 18 and ab = 72, what is (a − b)²? (a) 324 (b) 36 (c) 180 (d) 6
  10. 10.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 9) (b) (x + 4)(x + 6) (c) (x − 4)(x − 6) (d) (x + 1)(x + 24)
  11. 11.If a + b = 19 and ab = 84, what is a² + b²? (a) 361 (b) 529 (c) 277 (d) 193
  12. 12.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 4225 (b) 4235 (c) 4875 (d) 5625
  13. 13.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1681 (b) 1599 (c) 1521 (d) 1531
  14. 14.What is the sum of the roots of x² − 10x + 24 = 0? (a) 9 (b) 24 (c) 10 (d) 11
  15. 15.What is the product of the roots of x² − 14x + 48 = 0? (a) 48 (b) 14 (c) 47 (d) 49
  16. 16.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 5184 (b) 4896 (c) 4634 (d) 4624
  17. 17.What is the sum of the roots of x² − 11x + 24 = 0? (a) 10 (b) 24 (c) 12 (d) 11
  18. 18.Which is the correct factorisation of x² + 5x + 6? (a) (x − 2)(x − 3) (b) (x + 2)(x + 3) (c) (x + 1)(x + 6) (d) (x + 1)(x + 4)
  19. 19.Which is the correct factorisation of x² + 13x + 36? (a) (x − 4)(x − 9) (b) (x + 1)(x + 12) (c) (x + 4)(x + 9) (d) (x + 1)(x + 36)
  20. 20.Which is the correct factorisation of x² + 9x + 20? (a) (x + 4)(x + 5) (b) (x − 4)(x − 5) (c) (x + 1)(x + 8) (d) (x + 1)(x + 20)
  21. 21.If a + b = 15 and ab = 56, what is (a − b)²? (a) 225 (b) 1 (c) 169 (d) 113
  22. 22.If a + b = 13 and ab = 40, what is (a − b)²? (a) 3 (b) 9 (c) 169 (d) 89
  23. 23.What is the sum of the roots of x² − 14x + 48 = 0? (a) 13 (b) 14 (c) 15 (d) 48
  24. 24.What is the product of the roots of x² − 13x + 42 = 0? (a) 42 (b) 43 (c) 13 (d) 41
  25. 25.What are the roots of x² − 15x + 56 = 0? (a) 1 and 14 (b) 2 and 28 (c) 1 and 56 (d) 7 and 8
  26. 26.If a + b = 7 and ab = 10, what is a² + b²? (a) 29 (b) 49 (c) 39 (d) 69
  27. 27.If a + b = 8 and ab = 12, what is a² + b²? (a) 52 (b) 40 (c) 64 (d) 88
  28. 28.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 14) (b) (x + 1)(x + 54) (c) (x − 6)(x − 9) (d) (x + 6)(x + 9)
  29. 29.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3604 (b) 3844 (c) 3364 (d) 3854
  30. 30.If a + b = 4 and ab = 3, what is (a − b)²? (a) 16 (b) 4 (c) 2 (d) 10
  31. 31.If a + b = 19 and ab = 90, what is a² + b²? (a) 271 (b) 541 (c) 181 (d) 361
  32. 32.If a + b = 19 and ab = 84, what is (a − b)²? (a) 25 (b) 361 (c) 5 (d) 193
  33. 33.What is the product of the roots of x² − 11x + 28 = 0? (a) 27 (b) 28 (c) 29 (d) 11
  34. 34.Which is the correct factorisation of x² + 10x + 21? (a) (x − 3)(x − 7) (b) (x + 3)(x + 7) (c) (x + 1)(x + 21) (d) (x + 1)(x + 9)
  35. 35.If a + b = 18 and ab = 77, what is (a − b)²? (a) 324 (b) 4 (c) 16 (d) 170
  36. 36.If a + b = 20 and ab = 96, what is (a − b)²? (a) 400 (b) 4 (c) 208 (d) 16
  37. 37.If a + b = 8 and ab = 7, what is a² + b²? (a) 57 (b) 78 (c) 64 (d) 50
  38. 38.If a + b = 3 and ab = 2, what is a² + b²? (a) 7 (b) 5 (c) 9 (d) 13
  39. 39.What is the product of the roots of x² − 10x + 21 = 0? (a) 22 (b) 21 (c) 10 (d) 20
  40. 40.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 586 (b) 256 (c) 416 (d) 576

Answer key

  1. 34
  2. 16
  3. (x + 5)(x + 8)
  4. 6724
  5. 5184
  6. 25
  7. (x + 2)(x + 4)
  8. 30
  9. 36
  10. (x + 4)(x + 6)
  11. 193
  12. 4225
  13. 1521
  14. 10
  15. 48
  16. 4624
  17. 11
  18. (x + 2)(x + 3)
  19. (x + 4)(x + 9)
  20. (x + 4)(x + 5)
  21. 1
  22. 9
  23. 14
  24. 42
  25. 7 and 8
  26. 29
  27. 40
  28. (x + 6)(x + 9)
  29. 3844
  30. 4
  31. 181
  32. 25
  33. 28
  34. (x + 3)(x + 7)
  35. 16
  36. 16
  37. 50
  38. 5
  39. 21
  40. 576

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