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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 614 · 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 + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 8)(x + 9) (c) (x + 1)(x + 72) (d) (x + 1)(x + 16)
  2. 2.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x + 1)(x + 8) (c) (x + 4)(x + 5) (d) (x − 4)(x − 5)
  3. 3.If a + b = 14 and ab = 40, what is (a − b)²? (a) 196 (b) 6 (c) 116 (d) 36
  4. 4.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2304 (b) 2704 (c) 2314 (d) 2496
  5. 5.If a + b = 7 and ab = 12, what is (a − b)²? (a) 1 (b) 49 (c) 37 (d) 25
  6. 6.If a + b = 10 and ab = 21, what is (a − b)²? (a) 4 (b) 58 (c) 100 (d) 16
  7. 7.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8291 (b) 8281 (c) 7921 (d) 8101
  8. 8.What is the product of the roots of x² − 16x + 63 = 0? (a) 63 (b) 62 (c) 64 (d) 16
  9. 9.If a + b = 9 and ab = 14, what is a² + b²? (a) 81 (b) 67 (c) 109 (d) 53
  10. 10.If a + b = 4 and ab = 3, what is (a − b)²? (a) 10 (b) 4 (c) 2 (d) 16
  11. 11.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 5041 (b) 4761 (c) 5051 (d) 4901
  12. 12.What is the product of the roots of x² − 11x + 30 = 0? (a) 29 (b) 11 (c) 31 (d) 30
  13. 13.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1601 (b) 1681 (c) 1691 (d) 1521
  14. 14.If a + b = 20 and ab = 96, what is a² + b²? (a) 592 (b) 304 (c) 208 (d) 400
  15. 15.If a + b = 9 and ab = 18, what is (a − b)²? (a) 81 (b) 9 (c) 3 (d) 45
  16. 16.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x − 7)(x − 8) (c) (x + 1)(x + 14) (d) (x + 7)(x + 8)
  17. 17.If a + b = 6 and ab = 8, what is (a − b)²? (a) 20 (b) 4 (c) 36 (d) 2
  18. 18.What is the product of the roots of x² − 12x + 35 = 0? (a) 34 (b) 36 (c) 35 (d) 12
  19. 19.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 266 (b) 256 (c) 576 (d) 384
  20. 20.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 10) (b) (x − 2)(x − 5) (c) (x + 2)(x + 5) (d) (x + 1)(x + 6)
  21. 21.What are the roots of x² − 11x + 24 = 0? (a) 1 and 24 (b) 1 and 10 (c) 2 and 12 (d) 3 and 8
  22. 22.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)
  23. 23.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8659 (b) 8109 (c) 7569 (d) 8649
  24. 24.Which is the correct factorisation of x² + 13x + 40? (a) (x − 5)(x − 8) (b) (x + 1)(x + 40) (c) (x + 5)(x + 8) (d) (x + 1)(x + 12)
  25. 25.If a + b = 18 and ab = 80, what is a² + b²? (a) 244 (b) 324 (c) 484 (d) 164
  26. 26.What is the sum of the roots of x² − 12x + 35 = 0? (a) 12 (b) 11 (c) 35 (d) 13
  27. 27.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2516 (b) 2116 (c) 2916 (d) 2926
  28. 28.What is the sum of the roots of x² − 8x + 15 = 0? (a) 7 (b) 15 (c) 8 (d) 9
  29. 29.What is the product of the roots of x² − 12x + 32 = 0? (a) 33 (b) 12 (c) 31 (d) 32
  30. 30.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2611 (b) 2501 (c) 2601 (d) 2401
  31. 31.What is the sum of the roots of x² − 9x + 18 = 0? (a) 10 (b) 9 (c) 18 (d) 8
  32. 32.If a + b = 15 and ab = 56, what is (a − b)²? (a) 225 (b) 1 (c) 113 (d) 169
  33. 33.If a + b = 10 and ab = 16, what is a² + b²? (a) 100 (b) 132 (c) 84 (d) 68
  34. 34.What is the product of the roots of x² − 14x + 48 = 0? (a) 14 (b) 47 (c) 48 (d) 49
  35. 35.If a + b = 12 and ab = 32, what is (a − b)²? (a) 4 (b) 16 (c) 80 (d) 144
  36. 36.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1531 (b) 1681 (c) 1599 (d) 1521
  37. 37.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 10025 (b) 11025 (c) 9025 (d) 11035
  38. 38.Which is the correct factorisation of x² + 8x + 15? (a) (x − 3)(x − 5) (b) (x + 3)(x + 5) (c) (x + 1)(x + 15) (d) (x + 1)(x + 7)
  39. 39.If a + b = 13 and ab = 40, what is a² + b²? (a) 249 (b) 89 (c) 129 (d) 169
  40. 40.Which is the correct factorisation of x² + 13x + 42? (a) (x + 6)(x + 7) (b) (x + 1)(x + 42) (c) (x − 6)(x − 7) (d) (x + 1)(x + 12)

Answer key

  1. (x + 8)(x + 9)
  2. (x + 4)(x + 5)
  3. 36
  4. 2304
  5. 1
  6. 16
  7. 8281
  8. 63
  9. 53
  10. 4
  11. 5041
  12. 30
  13. 1681
  14. 208
  15. 9
  16. (x + 7)(x + 8)
  17. 4
  18. 35
  19. 256
  20. (x + 2)(x + 5)
  21. 3 and 8
  22. (x + 5)(x + 6)
  23. 8649
  24. (x + 5)(x + 8)
  25. 164
  26. 12
  27. 2916
  28. 8
  29. 32
  30. 2601
  31. 9
  32. 1
  33. 68
  34. 48
  35. 16
  36. 1521
  37. 11025
  38. (x + 3)(x + 5)
  39. 89
  40. (x + 6)(x + 7)

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