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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 285 · 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.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 2035 (b) 1625 (c) 1225 (d) 2025
  2. 2.What are the roots of x² − 16x + 63 = 0? (a) 1 and 63 (b) 3 and 21 (c) 1 and 15 (d) 7 and 9
  3. 3.If a + b = 18 and ab = 72, what is (a − b)²? (a) 324 (b) 180 (c) 6 (d) 36
  4. 4.What is the sum of the roots of x² − 10x + 16 = 0? (a) 9 (b) 11 (c) 16 (d) 10
  5. 5.If a + b = 19 and ab = 90, what is a² + b²? (a) 271 (b) 541 (c) 181 (d) 361
  6. 6.If a + b = 15 and ab = 56, what is (a − b)²? (a) 1 (b) 113 (c) 225 (d) 169
  7. 7.If a + b = 14 and ab = 40, what is (a − b)²? (a) 196 (b) 36 (c) 6 (d) 116
  8. 8.If a + b = 14 and ab = 24, what is (a − b)²? (a) 100 (b) 148 (c) 196 (d) 10
  9. 9.If a + b = 15 and ab = 50, what is (a − b)²? (a) 125 (b) 25 (c) 5 (d) 225
  10. 10.If a + b = 16 and ab = 55, what is a² + b²? (a) 256 (b) 366 (c) 146 (d) 201
  11. 11.If a + b = 8 and ab = 15, what is a² + b²? (a) 34 (b) 64 (c) 49 (d) 94
  12. 12.What is the product of the roots of x² − 14x + 45 = 0? (a) 46 (b) 44 (c) 14 (d) 45
  13. 13.If a + b = 9 and ab = 20, what is (a − b)²? (a) 41 (b) 1 (c) 81 (d) 61
  14. 14.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 625 (b) 235 (c) 375 (d) 225
  15. 15.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)
  16. 16.Which is the correct factorisation of x² + 9x + 14? (a) (x − 2)(x − 7) (b) (x + 2)(x + 7) (c) (x + 1)(x + 8) (d) (x + 1)(x + 14)
  17. 17.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 56) (c) (x + 1)(x + 14) (d) (x + 7)(x + 8)
  18. 18.What is the sum of the roots of x² − 11x + 30 = 0? (a) 12 (b) 11 (c) 10 (d) 30
  19. 19.If a + b = 10 and ab = 24, what is (a − b)²? (a) 2 (b) 52 (c) 4 (d) 100
  20. 20.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 9025 (b) 11025 (c) 10025 (d) 11035
  21. 21.What is the product of the roots of x² − 16x + 63 = 0? (a) 16 (b) 64 (c) 62 (d) 63
  22. 22.If a + b = 14 and ab = 45, what is (a − b)²? (a) 16 (b) 4 (c) 196 (d) 106
  23. 23.What is the product of the roots of x² − 11x + 28 = 0? (a) 11 (b) 29 (c) 27 (d) 28
  24. 24.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 7) (b) (x + 3)(x + 5) (c) (x − 3)(x − 5) (d) (x + 1)(x + 15)
  25. 25.If a + b = 3 and ab = 2, what is a² + b²? (a) 13 (b) 9 (c) 5 (d) 7
  26. 26.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x + 1)(x + 48) (c) (x − 6)(x − 8) (d) (x + 6)(x + 8)
  27. 27.If a + b = 17 and ab = 66, what is a² + b²? (a) 223 (b) 289 (c) 157 (d) 421
  28. 28.If a + b = 20 and ab = 96, what is (a − b)²? (a) 4 (b) 208 (c) 16 (d) 400
  29. 29.What are the roots of x² − 15x + 54 = 0? (a) 6 and 9 (b) 1 and 14 (c) 2 and 27 (d) 1 and 54
  30. 30.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 11) (b) (x − 3)(x − 9) (c) (x + 3)(x + 9) (d) (x + 1)(x + 27)
  31. 31.What are the roots of x² − 10x + 21 = 0? (a) 1 and 9 (b) 3 and 7 (c) 2 and 8 (d) 1 and 21
  32. 32.Which is the correct factorisation of x² + 8x + 12? (a) (x − 2)(x − 6) (b) (x + 2)(x + 6) (c) (x + 1)(x + 12) (d) (x + 1)(x + 7)
  33. 33.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 11) (b) (x − 4)(x − 8) (c) (x + 4)(x + 8) (d) (x + 1)(x + 32)
  34. 34.If a + b = 17 and ab = 72, what is a² + b²? (a) 145 (b) 433 (c) 217 (d) 289
  35. 35.If a + b = 7 and ab = 12, what is a² + b²? (a) 25 (b) 37 (c) 73 (d) 49
  36. 36.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 6889 (b) 5939 (c) 6391 (d) 5929
  37. 37.Which is the correct factorisation of x² + 11x + 30? (a) (x + 5)(x + 6) (b) (x + 1)(x + 30) (c) (x − 5)(x − 6) (d) (x + 1)(x + 10)
  38. 38.What are the roots of x² − 17x + 72 = 0? (a) 2 and 36 (b) 1 and 72 (c) 1 and 16 (d) 8 and 9
  39. 39.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)
  40. 40.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8101 (b) 8291 (c) 8281 (d) 7921

Answer key

  1. 2025
  2. 7 and 9
  3. 36
  4. 10
  5. 181
  6. 1
  7. 36
  8. 100
  9. 25
  10. 146
  11. 34
  12. 45
  13. 1
  14. 225
  15. (x + 8)(x + 9)
  16. (x + 2)(x + 7)
  17. (x + 7)(x + 8)
  18. 11
  19. 4
  20. 11025
  21. 63
  22. 16
  23. 28
  24. (x + 3)(x + 5)
  25. 5
  26. (x + 6)(x + 8)
  27. 157
  28. 16
  29. 6 and 9
  30. (x + 3)(x + 9)
  31. 3 and 7
  32. (x + 2)(x + 6)
  33. (x + 4)(x + 8)
  34. 145
  35. 25
  36. 5929
  37. (x + 5)(x + 6)
  38. 8 and 9
  39. (x + 5)(x + 8)
  40. 8281

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