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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 942 · 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 product of the roots of x² − 14x + 48 = 0? (a) 47 (b) 49 (c) 14 (d) 48
  2. 2.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x + 1)(x + 72) (c) (x + 1)(x + 16) (d) (x − 8)(x − 9)
  3. 3.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 1)(x + 48) (c) (x + 1)(x + 13) (d) (x + 6)(x + 8)
  4. 4.If a + b = 10 and ab = 24, what is (a − b)²? (a) 2 (b) 100 (c) 4 (d) 52
  5. 5.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 5041 (b) 4771 (c) 4761 (d) 4899
  6. 6.If a + b = 4 and ab = 3, what is a² + b²? (a) 16 (b) 22 (c) 10 (d) 13
  7. 7.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 18 (c) 16 (d) 17
  8. 8.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 884 (b) 686 (c) 676 (d) 1156
  9. 9.If a + b = 9 and ab = 18, what is a² + b²? (a) 117 (b) 63 (c) 45 (d) 81
  10. 10.If a + b = 19 and ab = 88, what is a² + b²? (a) 273 (b) 185 (c) 361 (d) 537
  11. 11.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x + 6)(x + 7) (c) (x + 1)(x + 42) (d) (x − 6)(x − 7)
  12. 12.What is the sum of the roots of x² − 10x + 24 = 0? (a) 24 (b) 11 (c) 10 (d) 9
  13. 13.If a + b = 20 and ab = 99, what is a² + b²? (a) 202 (b) 598 (c) 400 (d) 301
  14. 14.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 8649 (b) 7579 (c) 8091 (d) 7569
  15. 15.What are the roots of x² − 12x + 35 = 0? (a) 2 and 10 (b) 5 and 7 (c) 1 and 35 (d) 1 and 11
  16. 16.What are the roots of x² − 16x + 63 = 0? (a) 3 and 21 (b) 7 and 9 (c) 1 and 15 (d) 1 and 63
  17. 17.What are the roots of x² − 5x + 6 = 0? (a) 1 and 6 (b) 2 and 4 (c) 1 and 4 (d) 2 and 3
  18. 18.What is the sum of the roots of x² − 12x + 35 = 0? (a) 13 (b) 35 (c) 11 (d) 12
  19. 19.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x − 7)(x − 9) (c) (x + 7)(x + 9) (d) (x + 1)(x + 15)
  20. 20.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 6375 (b) 7225 (c) 5635 (d) 5625
  21. 21.If a + b = 14 and ab = 40, what is (a − b)²? (a) 196 (b) 36 (c) 6 (d) 116
  22. 22.What is the product of the roots of x² − 14x + 45 = 0? (a) 44 (b) 45 (c) 46 (d) 14
  23. 23.What is the sum of the roots of x² − 15x + 54 = 0? (a) 14 (b) 16 (c) 15 (d) 54
  24. 24.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 2525 (b) 3035 (c) 3025 (d) 2025
  25. 25.What is the sum of the roots of x² − 11x + 28 = 0? (a) 12 (b) 10 (c) 28 (d) 11
  26. 26.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 3025 (b) 4235 (c) 3625 (d) 4225
  27. 27.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 399 (b) 361 (c) 441 (d) 371
  28. 28.If a + b = 19 and ab = 90, what is (a − b)²? (a) 271 (b) 181 (c) 1 (d) 361
  29. 29.What is the product of the roots of x² − 13x + 36 = 0? (a) 13 (b) 37 (c) 36 (d) 35
  30. 30.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10004 (b) 10414 (c) 10404 (d) 9604
  31. 31.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 4489 (b) 5329 (c) 4909 (d) 5339
  32. 32.If a + b = 13 and ab = 30, what is a² + b²? (a) 109 (b) 169 (c) 139 (d) 229
  33. 33.If a + b = 12 and ab = 27, what is (a − b)²? (a) 90 (b) 6 (c) 144 (d) 36
  34. 34.If a + b = 18 and ab = 77, what is a² + b²? (a) 247 (b) 324 (c) 170 (d) 478
  35. 35.If a + b = 13 and ab = 42, what is a² + b²? (a) 253 (b) 169 (c) 127 (d) 85
  36. 36.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)
  37. 37.What is the product of the roots of x² − 10x + 24 = 0? (a) 23 (b) 10 (c) 24 (d) 25
  38. 38.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 28) (b) (x + 1)(x + 10) (c) (x − 4)(x − 7) (d) (x + 4)(x + 7)
  39. 39.If a + b = 21 and ab = 108, what is a² + b²? (a) 333 (b) 225 (c) 441 (d) 657
  40. 40.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6401 (b) 6561 (c) 6571 (d) 6241

Answer key

  1. 48
  2. (x + 8)(x + 9)
  3. (x + 6)(x + 8)
  4. 4
  5. 4761
  6. 10
  7. 17
  8. 676
  9. 45
  10. 185
  11. (x + 6)(x + 7)
  12. 10
  13. 202
  14. 7569
  15. 5 and 7
  16. 7 and 9
  17. 2 and 3
  18. 12
  19. (x + 7)(x + 9)
  20. 5625
  21. 36
  22. 45
  23. 15
  24. 3025
  25. 11
  26. 4225
  27. 361
  28. 1
  29. 36
  30. 10404
  31. 5329
  32. 109
  33. 36
  34. 170
  35. 85
  36. (x + 3)(x + 5)
  37. 24
  38. (x + 4)(x + 7)
  39. 225
  40. 6561

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