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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 249 · 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 = 7 and ab = 10, what is (a − b)²? (a) 49 (b) 9 (c) 3 (d) 29
  2. 2.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 6) (b) (x − 2)(x − 3) (c) (x + 2)(x + 3) (d) (x + 1)(x + 4)
  3. 3.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 8)(x + 9) (c) (x − 8)(x − 9) (d) (x + 1)(x + 16)
  4. 4.What is the product of the roots of x² − 16x + 63 = 0? (a) 62 (b) 16 (c) 63 (d) 64
  5. 5.If a + b = 12 and ab = 11, what is a² + b²? (a) 122 (b) 133 (c) 166 (d) 144
  6. 6.What is the sum of the roots of x² − 14x + 45 = 0? (a) 45 (b) 14 (c) 13 (d) 15
  7. 7.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 16) (b) (x − 2)(x − 8) (c) (x + 2)(x + 8) (d) (x + 1)(x + 9)
  8. 8.If a + b = 17 and ab = 70, what is (a − b)²? (a) 9 (b) 289 (c) 3 (d) 149
  9. 9.If a + b = 12 and ab = 20, what is a² + b²? (a) 124 (b) 104 (c) 184 (d) 144
  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 = 11 and ab = 28, what is (a − b)²? (a) 65 (b) 3 (c) 9 (d) 121
  12. 12.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 15 (c) 16 (d) 14
  13. 13.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 9216 (b) 10826 (c) 10016 (d) 10816
  14. 14.If a + b = 18 and ab = 72, what is a² + b²? (a) 180 (b) 324 (c) 468 (d) 252
  15. 15.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)
  16. 16.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 375 (b) 225 (c) 235 (d) 625
  17. 17.If a + b = 15 and ab = 50, what is a² + b²? (a) 225 (b) 125 (c) 325 (d) 175
  18. 18.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 539 (b) 289 (c) 529 (d) 409
  19. 19.What is the product of the roots of x² − 12x + 35 = 0? (a) 34 (b) 36 (c) 12 (d) 35
  20. 20.If a + b = 18 and ab = 80, what is a² + b²? (a) 164 (b) 484 (c) 244 (d) 324
  21. 21.What are the roots of x² − 6x + 8 = 0? (a) 1 and 8 (b) 2 and 5 (c) 1 and 5 (d) 2 and 4
  22. 22.If a + b = 13 and ab = 42, what is (a − b)²? (a) 127 (b) 1 (c) 85 (d) 169
  23. 23.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8109 (b) 7569 (c) 8659 (d) 8649
  24. 24.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 14) (c) (x + 1)(x + 56) (d) (x + 7)(x + 8)
  25. 25.Which is the correct factorisation of x² + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 5)(x + 6) (c) (x + 1)(x + 30) (d) (x + 1)(x + 10)
  26. 26.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 3136 (b) 4096 (c) 3146 (d) 3584
  27. 27.What is the product of the roots of x² − 11x + 24 = 0? (a) 24 (b) 23 (c) 11 (d) 25
  28. 28.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 8) (b) (x − 2)(x − 4) (c) (x + 1)(x + 5) (d) (x + 2)(x + 4)
  29. 29.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 9984 (b) 10816 (c) 9226 (d) 9216
  30. 30.If a + b = 11 and ab = 30, what is a² + b²? (a) 181 (b) 121 (c) 91 (d) 61
  31. 31.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 18) (b) (x − 3)(x − 6) (c) (x + 1)(x + 8) (d) (x + 3)(x + 6)
  32. 32.If a + b = 5 and ab = 4, what is (a − b)²? (a) 25 (b) 17 (c) 9 (d) 3
  33. 33.What is the sum of the roots of x² − 5x + 6 = 0? (a) 4 (b) 7 (c) 5 (d) 6
  34. 34.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)
  35. 35.If a + b = 12 and ab = 32, what is (a − b)²? (a) 144 (b) 16 (c) 80 (d) 4
  36. 36.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 10) (b) (x + 1)(x + 6) (c) (x − 2)(x − 5) (d) (x + 2)(x + 5)
  37. 37.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 7931 (b) 7921 (c) 8281 (d) 8099
  38. 38.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6401 (b) 6241 (c) 6561 (d) 6571
  39. 39.What is the product of the roots of x² − 14x + 48 = 0? (a) 48 (b) 47 (c) 14 (d) 49
  40. 40.If a + b = 14 and ab = 45, what is (a − b)²? (a) 4 (b) 106 (c) 196 (d) 16

Answer key

  1. 9
  2. (x + 2)(x + 3)
  3. (x + 8)(x + 9)
  4. 63
  5. 122
  6. 14
  7. (x + 2)(x + 8)
  8. 9
  9. 104
  10. (x + 4)(x + 6)
  11. 9
  12. 15
  13. 10816
  14. 180
  15. (x + 3)(x + 7)
  16. 225
  17. 125
  18. 529
  19. 35
  20. 164
  21. 2 and 4
  22. 1
  23. 8649
  24. (x + 7)(x + 8)
  25. (x + 5)(x + 6)
  26. 3136
  27. 24
  28. (x + 2)(x + 4)
  29. 9216
  30. 61
  31. (x + 3)(x + 6)
  32. 9
  33. 5
  34. (x + 4)(x + 8)
  35. 16
  36. (x + 2)(x + 5)
  37. 7921
  38. 6561
  39. 48
  40. 16

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