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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 568 · 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.What is the sum of the roots of x² − 15x + 56 = 0? (a) 14 (b) 15 (c) 16 (d) 56
  2. 2.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x + 1)(x + 16) (c) (x − 8)(x − 9) (d) (x + 1)(x + 72)
  3. 3.If a + b = 11 and ab = 28, what is (a − b)²? (a) 121 (b) 65 (c) 3 (d) 9
  4. 4.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 6)(x + 7) (c) (x + 1)(x + 12) (d) (x + 1)(x + 42)
  5. 5.What are the roots of x² − 5x + 6 = 0? (a) 1 and 4 (b) 2 and 4 (c) 1 and 6 (d) 2 and 3
  6. 6.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x − 7)(x − 8) (c) (x + 1)(x + 56) (d) (x + 7)(x + 8)
  7. 7.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 384 (b) 256 (c) 266 (d) 576
  8. 8.If a + b = 7 and ab = 10, what is a² + b²? (a) 69 (b) 39 (c) 49 (d) 29
  9. 9.If a + b = 9 and ab = 8, what is a² + b²? (a) 97 (b) 81 (c) 73 (d) 65
  10. 10.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 6)(x + 8) (c) (x + 1)(x + 13) (d) (x + 1)(x + 48)
  11. 11.If a + b = 20 and ab = 96, what is a² + b²? (a) 400 (b) 208 (c) 592 (d) 304
  12. 12.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2809 (b) 2219 (c) 2491 (d) 2209
  13. 13.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1604 (b) 1764 (c) 1444 (d) 1774
  14. 14.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2304 (b) 2714 (c) 2504 (d) 2704
  15. 15.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 9801 (b) 9999 (c) 10201 (d) 9811
  16. 16.Which is the correct factorisation of x² + 15x + 54? (a) (x + 6)(x + 9) (b) (x − 6)(x − 9) (c) (x + 1)(x + 54) (d) (x + 1)(x + 14)
  17. 17.If a + b = 16 and ab = 48, what is (a − b)²? (a) 8 (b) 64 (c) 160 (d) 256
  18. 18.If a + b = 19 and ab = 90, what is a² + b²? (a) 181 (b) 361 (c) 271 (d) 541
  19. 19.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6094 (b) 6084 (c) 6724 (d) 6396
  20. 20.If a + b = 10 and ab = 9, what is a² + b²? (a) 118 (b) 100 (c) 82 (d) 91
  21. 21.If a + b = 16 and ab = 60, what is a² + b²? (a) 376 (b) 136 (c) 256 (d) 196
  22. 22.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 4916 (b) 5486 (c) 4356 (d) 5476
  23. 23.If a + b = 11 and ab = 24, what is (a − b)²? (a) 121 (b) 25 (c) 5 (d) 73
  24. 24.If a + b = 6 and ab = 8, what is a² + b²? (a) 28 (b) 36 (c) 52 (d) 20
  25. 25.If a + b = 15 and ab = 36, what is (a − b)²? (a) 153 (b) 225 (c) 9 (d) 81
  26. 26.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 8846 (b) 7396 (c) 8836 (d) 8116
  27. 27.If a + b = 15 and ab = 56, what is (a − b)²? (a) 1 (b) 225 (c) 113 (d) 169
  28. 28.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 3025 (b) 2525 (c) 3035 (d) 2025
  29. 29.If a + b = 13 and ab = 12, what is a² + b²? (a) 157 (b) 145 (c) 169 (d) 193
  30. 30.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 8 and 9 (c) 1 and 16 (d) 2 and 36
  31. 31.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 8125 (b) 7225 (c) 9025 (d) 9035
  32. 32.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3591 (b) 3259 (c) 3969 (d) 3249
  33. 33.Which is the correct factorisation of x² + 8x + 15? (a) (x − 3)(x − 5) (b) (x + 3)(x + 5) (c) (x + 1)(x + 15) (d) (x + 1)(x + 7)
  34. 34.If a + b = 20 and ab = 99, what is (a − b)²? (a) 4 (b) 2 (c) 202 (d) 400
  35. 35.What is the product of the roots of x² − 14x + 45 = 0? (a) 44 (b) 46 (c) 45 (d) 14
  36. 36.What is the product of the roots of x² − 10x + 24 = 0? (a) 10 (b) 23 (c) 24 (d) 25
  37. 37.If a + b = 17 and ab = 70, what is a² + b²? (a) 289 (b) 429 (c) 219 (d) 149
  38. 38.Which is the correct factorisation of x² + 12x + 32? (a) (x + 4)(x + 8) (b) (x + 1)(x + 11) (c) (x − 4)(x − 8) (d) (x + 1)(x + 32)
  39. 39.What are the roots of x² − 9x + 20 = 0? (a) 2 and 10 (b) 1 and 8 (c) 1 and 20 (d) 4 and 5
  40. 40.What is the product of the roots of x² − 13x + 42 = 0? (a) 42 (b) 43 (c) 13 (d) 41

Answer key

  1. 15
  2. (x + 8)(x + 9)
  3. 9
  4. (x + 6)(x + 7)
  5. 2 and 3
  6. (x + 7)(x + 8)
  7. 256
  8. 29
  9. 65
  10. (x + 6)(x + 8)
  11. 208
  12. 2209
  13. 1764
  14. 2704
  15. 9801
  16. (x + 6)(x + 9)
  17. 64
  18. 181
  19. 6084
  20. 82
  21. 136
  22. 5476
  23. 25
  24. 20
  25. 81
  26. 8836
  27. 1
  28. 3025
  29. 145
  30. 8 and 9
  31. 9025
  32. 3249
  33. (x + 3)(x + 5)
  34. 4
  35. 45
  36. 24
  37. 149
  38. (x + 4)(x + 8)
  39. 4 and 5
  40. 42

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