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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 386 · 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 + 10x + 16? (a) (x − 2)(x − 8) (b) (x + 2)(x + 8) (c) (x + 1)(x + 16) (d) (x + 1)(x + 9)
  2. 2.What are the roots of x² − 13x + 40 = 0? (a) 1 and 40 (b) 1 and 12 (c) 2 and 20 (d) 5 and 8
  3. 3.What are the roots of x² − 16x + 63 = 0? (a) 1 and 15 (b) 1 and 63 (c) 3 and 21 (d) 7 and 9
  4. 4.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 7)(x + 9) (c) (x − 7)(x − 9) (d) (x + 1)(x + 63)
  5. 5.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 8) (b) (x + 2)(x + 4) (c) (x − 2)(x − 4) (d) (x + 1)(x + 5)
  6. 6.What is the sum of the roots of x² − 8x + 12 = 0? (a) 8 (b) 12 (c) 9 (d) 7
  7. 7.If a + b = 18 and ab = 80, what is a² + b²? (a) 164 (b) 484 (c) 244 (d) 324
  8. 8.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 17 (c) 72 (d) 16
  9. 9.If a + b = 20 and ab = 96, what is a² + b²? (a) 400 (b) 304 (c) 592 (d) 208
  10. 10.If a + b = 7 and ab = 6, what is (a − b)²? (a) 5 (b) 25 (c) 37 (d) 49
  11. 11.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 576 (b) 256 (c) 586 (d) 416
  12. 12.What is the product of the roots of x² − 15x + 56 = 0? (a) 57 (b) 56 (c) 15 (d) 55
  13. 13.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 1156 (b) 676 (c) 686 (d) 884
  14. 14.What is the sum of the roots of x² − 14x + 48 = 0? (a) 48 (b) 15 (c) 13 (d) 14
  15. 15.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 8) (b) (x + 1)(x + 18) (c) (x − 3)(x − 6) (d) (x + 3)(x + 6)
  16. 16.If a + b = 13 and ab = 22, what is (a − b)²? (a) 9 (b) 169 (c) 81 (d) 125
  17. 17.If a + b = 19 and ab = 90, what is a² + b²? (a) 361 (b) 271 (c) 541 (d) 181
  18. 18.If a + b = 11 and ab = 18, what is a² + b²? (a) 121 (b) 85 (c) 157 (d) 103
  19. 19.What is the product of the roots of x² − 10x + 16 = 0? (a) 16 (b) 10 (c) 17 (d) 15
  20. 20.What are the roots of x² − 11x + 18 = 0? (a) 3 and 6 (b) 2 and 9 (c) 1 and 18 (d) 1 and 10
  21. 21.If a + b = 9 and ab = 14, what is (a − b)²? (a) 81 (b) 25 (c) 5 (d) 53
  22. 22.What are the roots of x² − 11x + 28 = 0? (a) 1 and 28 (b) 1 and 10 (c) 2 and 14 (d) 4 and 7
  23. 23.If a + b = 15 and ab = 50, what is a² + b²? (a) 225 (b) 325 (c) 125 (d) 175
  24. 24.If a + b = 9 and ab = 20, what is a² + b²? (a) 121 (b) 61 (c) 81 (d) 41
  25. 25.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3259 (b) 3249 (c) 3969 (d) 3591
  26. 26.What is the product of the roots of x² − 7x + 10 = 0? (a) 10 (b) 11 (c) 9 (d) 7
  27. 27.If a + b = 9 and ab = 14, what is a² + b²? (a) 81 (b) 109 (c) 53 (d) 67
  28. 28.If a + b = 15 and ab = 44, what is (a − b)²? (a) 7 (b) 49 (c) 137 (d) 225
  29. 29.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3731 (b) 3481 (c) 3601 (d) 3721
  30. 30.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 4901 (b) 5041 (c) 5051 (d) 4761
  31. 31.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2504 (b) 2304 (c) 2714 (d) 2704
  32. 32.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 7235 (b) 8075 (c) 7225 (d) 9025
  33. 33.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 1 and 16 (c) 8 and 9 (d) 2 and 36
  34. 34.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 635 (b) 425 (c) 625 (d) 225
  35. 35.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 1)(x + 16) (c) (x − 8)(x − 9) (d) (x + 8)(x + 9)
  36. 36.What is the product of the roots of x² − 12x + 32 = 0? (a) 31 (b) 32 (c) 12 (d) 33
  37. 37.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9025 (b) 9035 (c) 11025 (d) 9975
  38. 38.If a + b = 18 and ab = 72, what is a² + b²? (a) 324 (b) 252 (c) 468 (d) 180
  39. 39.What is the product of the roots of x² − 12x + 35 = 0? (a) 34 (b) 35 (c) 36 (d) 12
  40. 40.What are the roots of x² − 12x + 27 = 0? (a) 2 and 10 (b) 1 and 11 (c) 1 and 27 (d) 3 and 9

Answer key

  1. (x + 2)(x + 8)
  2. 5 and 8
  3. 7 and 9
  4. (x + 7)(x + 9)
  5. (x + 2)(x + 4)
  6. 8
  7. 164
  8. 17
  9. 208
  10. 25
  11. 576
  12. 56
  13. 676
  14. 14
  15. (x + 3)(x + 6)
  16. 81
  17. 181
  18. 85
  19. 16
  20. 2 and 9
  21. 25
  22. 4 and 7
  23. 125
  24. 41
  25. 3249
  26. 10
  27. 53
  28. 49
  29. 3721
  30. 5041
  31. 2704
  32. 7225
  33. 8 and 9
  34. 625
  35. (x + 8)(x + 9)
  36. 32
  37. 9025
  38. 180
  39. 35
  40. 3 and 9

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