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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 228 · 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 = 19 and ab = 84, what is a² + b²? (a) 277 (b) 361 (c) 193 (d) 529
  2. 2.What are the roots of x² − 13x + 42 = 0? (a) 1 and 42 (b) 6 and 7 (c) 2 and 21 (d) 1 and 12
  3. 3.What is the sum of the roots of x² − 17x + 72 = 0? (a) 17 (b) 72 (c) 16 (d) 18
  4. 4.Which is the correct factorisation of x² + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 1)(x + 6) (c) (x + 1)(x + 12) (d) (x + 3)(x + 4)
  5. 5.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 1024 (b) 784 (c) 904 (d) 1034
  6. 6.If a + b = 20 and ab = 99, what is a² + b²? (a) 400 (b) 202 (c) 598 (d) 301
  7. 7.If a + b = 11 and ab = 24, what is (a − b)²? (a) 73 (b) 5 (c) 25 (d) 121
  8. 8.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3591 (b) 3969 (c) 3259 (d) 3249
  9. 9.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 4225 (b) 5625 (c) 4875 (d) 4235
  10. 10.If a + b = 21 and ab = 108, what is (a − b)²? (a) 225 (b) 441 (c) 3 (d) 9
  11. 11.Which is the correct factorisation of x² + 11x + 30? (a) (x + 5)(x + 6) (b) (x + 1)(x + 30) (c) (x − 5)(x − 6) (d) (x + 1)(x + 10)
  12. 12.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 8) (b) (x + 1)(x + 14) (c) (x − 2)(x − 7) (d) (x + 2)(x + 7)
  13. 13.If a + b = 5 and ab = 4, what is a² + b²? (a) 17 (b) 33 (c) 21 (d) 25
  14. 14.If a + b = 20 and ab = 99, what is (a − b)²? (a) 202 (b) 2 (c) 400 (d) 4
  15. 15.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)
  16. 16.If a + b = 14 and ab = 24, what is a² + b²? (a) 172 (b) 244 (c) 148 (d) 196
  17. 17.What is the sum of the roots of x² − 9x + 18 = 0? (a) 10 (b) 8 (c) 9 (d) 18
  18. 18.If a + b = 6 and ab = 8, what is a² + b²? (a) 36 (b) 28 (c) 52 (d) 20
  19. 19.What are the roots of x² − 7x + 10 = 0? (a) 1 and 10 (b) 3 and 4 (c) 2 and 5 (d) 1 and 6
  20. 20.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 63) (c) (x + 7)(x + 9) (d) (x + 1)(x + 15)
  21. 21.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4771 (b) 5041 (c) 4899 (d) 4761
  22. 22.If a + b = 8 and ab = 7, what is (a − b)²? (a) 64 (b) 50 (c) 6 (d) 36
  23. 23.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 256 (b) 416 (c) 586 (d) 576
  24. 24.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 27) (b) (x + 3)(x + 9) (c) (x + 1)(x + 11) (d) (x − 3)(x − 9)
  25. 25.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 9409 (b) 10609 (c) 9419 (d) 9991
  26. 26.What are the roots of x² − 8x + 15 = 0? (a) 1 and 7 (b) 2 and 6 (c) 3 and 5 (d) 1 and 15
  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.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 10) (b) (x + 1)(x + 6) (c) (x − 2)(x − 5) (d) (x + 2)(x + 5)
  29. 29.What are the roots of x² − 6x + 8 = 0? (a) 1 and 8 (b) 2 and 5 (c) 2 and 4 (d) 1 and 5
  30. 30.If a + b = 11 and ab = 10, what is a² + b²? (a) 141 (b) 101 (c) 121 (d) 111
  31. 31.If a + b = 17 and ab = 72, what is (a − b)²? (a) 217 (b) 289 (c) 145 (d) 1
  32. 32.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1849 (b) 1379 (c) 1369 (d) 1591
  33. 33.What is the sum of the roots of x² − 10x + 16 = 0? (a) 10 (b) 11 (c) 9 (d) 16
  34. 34.What is the sum of the roots of x² − 16x + 63 = 0? (a) 16 (b) 15 (c) 17 (d) 63
  35. 35.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 3575 (b) 4225 (c) 3035 (d) 3025
  36. 36.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2501 (b) 2611 (c) 2401 (d) 2601
  37. 37.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1609 (b) 1369 (c) 1859 (d) 1849
  38. 38.What is the product of the roots of x² − 12x + 35 = 0? (a) 36 (b) 34 (c) 35 (d) 12
  39. 39.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8101 (b) 8281 (c) 7921 (d) 8291
  40. 40.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 8) (b) (x + 1)(x + 20) (c) (x − 4)(x − 5) (d) (x + 4)(x + 5)

Answer key

  1. 193
  2. 6 and 7
  3. 17
  4. (x + 3)(x + 4)
  5. 1024
  6. 202
  7. 25
  8. 3249
  9. 4225
  10. 9
  11. (x + 5)(x + 6)
  12. (x + 2)(x + 7)
  13. 17
  14. 4
  15. (x + 7)(x + 8)
  16. 148
  17. 9
  18. 20
  19. 2 and 5
  20. (x + 7)(x + 9)
  21. 4761
  22. 36
  23. 576
  24. (x + 3)(x + 9)
  25. 9409
  26. 3 and 5
  27. (x + 8)(x + 9)
  28. (x + 2)(x + 5)
  29. 2 and 4
  30. 101
  31. 1
  32. 1369
  33. 10
  34. 16
  35. 3025
  36. 2601
  37. 1849
  38. 35
  39. 8281
  40. (x + 4)(x + 5)

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