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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 438 · 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 19²? (a) 399 (b) 361 (c) 371 (d) 441
  2. 2.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 16 (c) 72 (d) 17
  3. 3.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)
  4. 4.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6241 (b) 6401 (c) 6571 (d) 6561
  5. 5.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)
  6. 6.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 451 (b) 361 (c) 401 (d) 441
  7. 7.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 1)(x + 63) (c) (x + 7)(x + 9) (d) (x − 7)(x − 9)
  8. 8.If a + b = 14 and ab = 40, what is (a − b)²? (a) 196 (b) 36 (c) 116 (d) 6
  9. 9.If a + b = 12 and ab = 27, what is a² + b²? (a) 117 (b) 90 (c) 144 (d) 198
  10. 10.If a + b = 19 and ab = 90, what is a² + b²? (a) 181 (b) 541 (c) 361 (d) 271
  11. 11.What is the product of the roots of x² − 15x + 54 = 0? (a) 53 (b) 55 (c) 15 (d) 54
  12. 12.If a + b = 18 and ab = 80, what is a² + b²? (a) 484 (b) 324 (c) 164 (d) 244
  13. 13.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 4916 (b) 4356 (c) 5486 (d) 5476
  14. 14.If a + b = 12 and ab = 32, what is a² + b²? (a) 208 (b) 80 (c) 112 (d) 144
  15. 15.If a + b = 11 and ab = 24, what is a² + b²? (a) 97 (b) 169 (c) 73 (d) 121
  16. 16.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 1)(x + 14) (c) (x − 7)(x − 8) (d) (x + 7)(x + 8)
  17. 17.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 12) (b) (x − 2)(x − 6) (c) (x + 2)(x + 6) (d) (x + 1)(x + 7)
  18. 18.If a + b = 11 and ab = 28, what is a² + b²? (a) 121 (b) 93 (c) 177 (d) 65
  19. 19.If a + b = 8 and ab = 7, what is a² + b²? (a) 64 (b) 50 (c) 78 (d) 57
  20. 20.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)
  21. 21.If a + b = 20 and ab = 99, what is a² + b²? (a) 400 (b) 202 (c) 301 (d) 598
  22. 22.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 425 (b) 225 (c) 635 (d) 625
  23. 23.If a + b = 11 and ab = 18, what is (a − b)²? (a) 7 (b) 49 (c) 121 (d) 85
  24. 24.If a + b = 13 and ab = 40, what is a² + b²? (a) 249 (b) 169 (c) 129 (d) 89
  25. 25.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 891 (b) 739 (c) 729 (d) 1089
  26. 26.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x + 1)(x + 8) (c) (x − 4)(x − 5) (d) (x + 4)(x + 5)
  27. 27.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)
  28. 28.If a + b = 20 and ab = 99, what is (a − b)²? (a) 4 (b) 2 (c) 400 (d) 202
  29. 29.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 3025 (b) 2525 (c) 2025 (d) 3035
  30. 30.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 5776 (b) 7056 (c) 5786 (d) 6384
  31. 31.If a + b = 11 and ab = 24, what is (a − b)²? (a) 73 (b) 121 (c) 5 (d) 25
  32. 32.What is the product of the roots of x² − 7x + 12 = 0? (a) 7 (b) 13 (c) 12 (d) 11
  33. 33.If a + b = 17 and ab = 70, what is a² + b²? (a) 149 (b) 289 (c) 429 (d) 219
  34. 34.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 10) (b) (x − 4)(x − 7) (c) (x + 4)(x + 7) (d) (x + 1)(x + 28)
  35. 35.If a + b = 16 and ab = 63, what is (a − b)²? (a) 130 (b) 4 (c) 256 (d) 2
  36. 36.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 18) (b) (x − 2)(x − 9) (c) (x + 2)(x + 9) (d) (x + 1)(x + 10)
  37. 37.What are the roots of x² − 14x + 48 = 0? (a) 6 and 8 (b) 1 and 48 (c) 2 and 24 (d) 1 and 13
  38. 38.If a + b = 15 and ab = 50, what is a² + b²? (a) 125 (b) 175 (c) 325 (d) 225
  39. 39.Which is the correct factorisation of x² + 8x + 15? (a) (x − 3)(x − 5) (b) (x + 1)(x + 7) (c) (x + 3)(x + 5) (d) (x + 1)(x + 15)
  40. 40.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 409 (b) 539 (c) 289 (d) 529

Answer key

  1. 361
  2. 17
  3. (x + 5)(x + 6)
  4. 6561
  5. (x + 2)(x + 3)
  6. 441
  7. (x + 7)(x + 9)
  8. 36
  9. 90
  10. 181
  11. 54
  12. 164
  13. 5476
  14. 80
  15. 73
  16. (x + 7)(x + 8)
  17. (x + 2)(x + 6)
  18. 65
  19. 50
  20. (x + 4)(x + 6)
  21. 202
  22. 625
  23. 49
  24. 89
  25. 729
  26. (x + 4)(x + 5)
  27. (x + 8)(x + 9)
  28. 4
  29. 3025
  30. 5776
  31. 25
  32. 12
  33. 149
  34. (x + 4)(x + 7)
  35. 4
  36. (x + 2)(x + 9)
  37. 6 and 8
  38. 125
  39. (x + 3)(x + 5)
  40. 529

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