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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 682 · 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 = 16 and ab = 63, what is a² + b²? (a) 193 (b) 130 (c) 382 (d) 256
  2. 2.What is the product of the roots of x² − 9x + 18 = 0? (a) 17 (b) 19 (c) 9 (d) 18
  3. 3.What is the product of the roots of x² − 15x + 56 = 0? (a) 55 (b) 15 (c) 56 (d) 57
  4. 4.What are the roots of x² − 11x + 30 = 0? (a) 1 and 10 (b) 1 and 30 (c) 2 and 15 (d) 5 and 6
  5. 5.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 45) (b) (x + 1)(x + 13) (c) (x − 5)(x − 9) (d) (x + 5)(x + 9)
  6. 6.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 9801 (b) 9999 (c) 10201 (d) 9811
  7. 7.If a + b = 16 and ab = 60, what is (a − b)²? (a) 256 (b) 16 (c) 136 (d) 4
  8. 8.If a + b = 21 and ab = 108, what is (a − b)²? (a) 225 (b) 9 (c) 441 (d) 3
  9. 9.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x + 1)(x + 14) (c) (x − 6)(x − 9) (d) (x + 6)(x + 9)
  10. 10.If a + b = 5 and ab = 6, what is (a − b)²? (a) 19 (b) 13 (c) 1 (d) 25
  11. 11.What are the roots of x² − 12x + 35 = 0? (a) 1 and 11 (b) 1 and 35 (c) 2 and 10 (d) 5 and 7
  12. 12.If a + b = 14 and ab = 45, what is a² + b²? (a) 286 (b) 151 (c) 106 (d) 196
  13. 13.If a + b = 9 and ab = 20, what is a² + b²? (a) 41 (b) 61 (c) 121 (d) 81
  14. 14.If a + b = 15 and ab = 56, what is a² + b²? (a) 337 (b) 113 (c) 169 (d) 225
  15. 15.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 6) (b) (x − 3)(x − 4) (c) (x + 3)(x + 4) (d) (x + 1)(x + 12)
  16. 16.What are the roots of x² − 14x + 48 = 0? (a) 1 and 48 (b) 2 and 24 (c) 6 and 8 (d) 1 and 13
  17. 17.What are the roots of x² − 9x + 14 = 0? (a) 3 and 6 (b) 1 and 14 (c) 2 and 7 (d) 1 and 8
  18. 18.What are the roots of x² − 9x + 20 = 0? (a) 2 and 10 (b) 4 and 5 (c) 1 and 20 (d) 1 and 8
  19. 19.What is the product of the roots of x² − 10x + 16 = 0? (a) 15 (b) 17 (c) 16 (d) 10
  20. 20.If a + b = 7 and ab = 6, what is (a − b)²? (a) 5 (b) 49 (c) 25 (d) 37
  21. 21.What is the sum of the roots of x² − 9x + 18 = 0? (a) 18 (b) 10 (c) 8 (d) 9
  22. 22.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2209 (b) 2491 (c) 2219 (d) 2809
  23. 23.If a + b = 17 and ab = 72, what is a² + b²? (a) 433 (b) 217 (c) 289 (d) 145
  24. 24.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 1)(x + 72) (c) (x + 8)(x + 9) (d) (x − 8)(x − 9)
  25. 25.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 10826 (b) 9216 (c) 10016 (d) 10816
  26. 26.What are the roots of x² − 12x + 27 = 0? (a) 2 and 10 (b) 1 and 11 (c) 3 and 9 (d) 1 and 27
  27. 27.If a + b = 11 and ab = 24, what is (a − b)²? (a) 121 (b) 73 (c) 25 (d) 5
  28. 28.If a + b = 20 and ab = 96, what is (a − b)²? (a) 400 (b) 16 (c) 4 (d) 208
  29. 29.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 399 (b) 371 (c) 441 (d) 361
  30. 30.What is the sum of the roots of x² − 11x + 30 = 0? (a) 10 (b) 11 (c) 30 (d) 12
  31. 31.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 15) (b) (x − 3)(x − 5) (c) (x + 3)(x + 5) (d) (x + 1)(x + 7)
  32. 32.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 6384 (b) 5776 (c) 5786 (d) 7056
  33. 33.If a + b = 19 and ab = 88, what is (a − b)²? (a) 3 (b) 185 (c) 9 (d) 361
  34. 34.What is the product of the roots of x² − 11x + 18 = 0? (a) 18 (b) 19 (c) 11 (d) 17
  35. 35.What are the roots of x² − 15x + 56 = 0? (a) 1 and 14 (b) 1 and 56 (c) 7 and 8 (d) 2 and 28
  36. 36.What is the product of the roots of x² − 9x + 20 = 0? (a) 21 (b) 19 (c) 20 (d) 9
  37. 37.If a + b = 18 and ab = 72, what is (a − b)²? (a) 180 (b) 324 (c) 6 (d) 36
  38. 38.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 6425 (b) 5625 (c) 7225 (d) 7235
  39. 39.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 12) (b) (x + 1)(x + 40) (c) (x − 5)(x − 8) (d) (x + 5)(x + 8)
  40. 40.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10404 (b) 10004 (c) 9604 (d) 10414

Answer key

  1. 130
  2. 18
  3. 56
  4. 5 and 6
  5. (x + 5)(x + 9)
  6. 9801
  7. 16
  8. 9
  9. (x + 6)(x + 9)
  10. 1
  11. 5 and 7
  12. 106
  13. 41
  14. 113
  15. (x + 3)(x + 4)
  16. 6 and 8
  17. 2 and 7
  18. 4 and 5
  19. 16
  20. 25
  21. 9
  22. 2209
  23. 145
  24. (x + 8)(x + 9)
  25. 10816
  26. 3 and 9
  27. 25
  28. 16
  29. 361
  30. 11
  31. (x + 3)(x + 5)
  32. 5776
  33. 9
  34. 18
  35. 7 and 8
  36. 20
  37. 36
  38. 7225
  39. (x + 5)(x + 8)
  40. 10404

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