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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 900 · 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 = 19 and ab = 84, what is (a − b)²? (a) 25 (b) 361 (c) 5 (d) 193
  2. 2.If a + b = 9 and ab = 18, what is a² + b²? (a) 63 (b) 81 (c) 117 (d) 45
  3. 3.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x − 5)(x − 6) (c) (x + 1)(x + 10) (d) (x + 5)(x + 6)
  4. 4.If a + b = 15 and ab = 56, what is (a − b)²? (a) 225 (b) 169 (c) 113 (d) 1
  5. 5.Which is the correct factorisation of x² + 12x + 32? (a) (x + 4)(x + 8) (b) (x + 1)(x + 32) (c) (x + 1)(x + 11) (d) (x − 4)(x − 8)
  6. 6.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x + 1)(x + 35) (c) (x − 5)(x − 7) (d) (x + 5)(x + 7)
  7. 7.If a + b = 15 and ab = 44, what is a² + b²? (a) 225 (b) 137 (c) 313 (d) 181
  8. 8.What are the roots of x² − 12x + 27 = 0? (a) 1 and 27 (b) 1 and 11 (c) 3 and 9 (d) 2 and 10
  9. 9.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 9216 (b) 10816 (c) 10016 (d) 10826
  10. 10.What are the roots of x² − 15x + 56 = 0? (a) 1 and 56 (b) 1 and 14 (c) 7 and 8 (d) 2 and 28
  11. 11.If a + b = 11 and ab = 10, what is a² + b²? (a) 101 (b) 121 (c) 141 (d) 111
  12. 12.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 10) (b) (x + 1)(x + 28) (c) (x − 4)(x − 7) (d) (x + 4)(x + 7)
  13. 13.What is the sum of the roots of x² − 13x + 40 = 0? (a) 12 (b) 13 (c) 40 (d) 14
  14. 14.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 6) (b) (x + 2)(x + 5) (c) (x − 2)(x − 5) (d) (x + 1)(x + 10)
  15. 15.If a + b = 21 and ab = 108, what is (a − b)²? (a) 441 (b) 225 (c) 9 (d) 3
  16. 16.What are the roots of x² − 15x + 54 = 0? (a) 1 and 14 (b) 1 and 54 (c) 2 and 27 (d) 6 and 9
  17. 17.What is the product of the roots of x² − 14x + 48 = 0? (a) 49 (b) 14 (c) 48 (d) 47
  18. 18.If a + b = 19 and ab = 90, what is (a − b)²? (a) 1 (b) 271 (c) 361 (d) 181
  19. 19.What are the roots of x² − 6x + 8 = 0? (a) 2 and 5 (b) 1 and 5 (c) 2 and 4 (d) 1 and 8
  20. 20.If a + b = 15 and ab = 54, what is (a − b)²? (a) 9 (b) 117 (c) 3 (d) 225
  21. 21.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 5939 (b) 5929 (c) 6889 (d) 6391
  22. 22.If a + b = 11 and ab = 30, what is a² + b²? (a) 121 (b) 61 (c) 91 (d) 181
  23. 23.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 8) (b) (x + 1)(x + 5) (c) (x − 2)(x − 4) (d) (x + 2)(x + 4)
  24. 24.If a + b = 8 and ab = 15, what is a² + b²? (a) 64 (b) 34 (c) 49 (d) 94
  25. 25.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 256 (b) 586 (c) 576 (d) 416
  26. 26.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10211 (b) 10001 (c) 10201 (d) 9801
  27. 27.If a + b = 12 and ab = 27, what is a² + b²? (a) 117 (b) 198 (c) 144 (d) 90
  28. 28.What is the product of the roots of x² − 13x + 42 = 0? (a) 13 (b) 42 (c) 43 (d) 41
  29. 29.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 625 (b) 225 (c) 425 (d) 635
  30. 30.What is the sum of the roots of x² − 12x + 35 = 0? (a) 12 (b) 11 (c) 35 (d) 13
  31. 31.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 16 (c) 17 (d) 72
  32. 32.If a + b = 15 and ab = 36, what is (a − b)²? (a) 9 (b) 225 (c) 81 (d) 153
  33. 33.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)
  34. 34.What are the roots of x² − 13x + 42 = 0? (a) 1 and 42 (b) 1 and 12 (c) 6 and 7 (d) 2 and 21
  35. 35.If a + b = 20 and ab = 99, what is (a − b)²? (a) 2 (b) 202 (c) 4 (d) 400
  36. 36.If a + b = 8 and ab = 7, what is a² + b²? (a) 64 (b) 78 (c) 57 (d) 50
  37. 37.What is the product of the roots of x² − 11x + 24 = 0? (a) 11 (b) 24 (c) 25 (d) 23
  38. 38.What is the product of the roots of x² − 12x + 32 = 0? (a) 33 (b) 12 (c) 32 (d) 31
  39. 39.If a + b = 10 and ab = 21, what is a² + b²? (a) 100 (b) 79 (c) 142 (d) 58
  40. 40.Which is the correct factorisation of x² + 10x + 16? (a) (x − 2)(x − 8) (b) (x + 1)(x + 16) (c) (x + 1)(x + 9) (d) (x + 2)(x + 8)

Answer key

  1. 25
  2. 45
  3. (x + 5)(x + 6)
  4. 1
  5. (x + 4)(x + 8)
  6. (x + 5)(x + 7)
  7. 137
  8. 3 and 9
  9. 10816
  10. 7 and 8
  11. 101
  12. (x + 4)(x + 7)
  13. 13
  14. (x + 2)(x + 5)
  15. 9
  16. 6 and 9
  17. 48
  18. 1
  19. 2 and 4
  20. 9
  21. 5929
  22. 61
  23. (x + 2)(x + 4)
  24. 34
  25. 576
  26. 10201
  27. 90
  28. 42
  29. 625
  30. 12
  31. 17
  32. 81
  33. (x + 5)(x + 8)
  34. 6 and 7
  35. 4
  36. 50
  37. 24
  38. 32
  39. 58
  40. (x + 2)(x + 8)

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