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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 157 · 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 − 9x + 18 = 0? (a) 9 (b) 18 (c) 17 (d) 19
  2. 2.What is the product of the roots of x² − 16x + 63 = 0? (a) 64 (b) 16 (c) 63 (d) 62
  3. 3.What are the roots of x² − 11x + 30 = 0? (a) 2 and 15 (b) 1 and 10 (c) 1 and 30 (d) 5 and 6
  4. 4.What is the sum of the roots of x² − 16x + 63 = 0? (a) 15 (b) 16 (c) 63 (d) 17
  5. 5.What is the product of the roots of x² − 10x + 16 = 0? (a) 17 (b) 10 (c) 15 (d) 16
  6. 6.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 896 (b) 1024 (c) 784 (d) 794
  7. 7.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)
  8. 8.If a + b = 16 and ab = 63, what is a² + b²? (a) 256 (b) 382 (c) 193 (d) 130
  9. 9.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 4) (b) (x + 2)(x + 3) (c) (x − 2)(x − 3) (d) (x + 1)(x + 6)
  10. 10.If a + b = 18 and ab = 72, what is a² + b²? (a) 468 (b) 180 (c) 252 (d) 324
  11. 11.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2304 (b) 2714 (c) 2704 (d) 2504
  12. 12.If a + b = 5 and ab = 6, what is a² + b²? (a) 19 (b) 25 (c) 37 (d) 13
  13. 13.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 10) (b) (x + 2)(x + 9) (c) (x + 1)(x + 18) (d) (x − 2)(x − 9)
  14. 14.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 72 (c) 71 (d) 17
  15. 15.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 3025 (b) 4235 (c) 3625 (d) 4225
  16. 16.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x + 1)(x + 56) (c) (x + 1)(x + 14) (d) (x − 7)(x − 8)
  17. 17.What is the product of the roots of x² − 15x + 54 = 0? (a) 54 (b) 15 (c) 53 (d) 55
  18. 18.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 9) (b) (x + 2)(x + 8) (c) (x − 2)(x − 8) (d) (x + 1)(x + 16)
  19. 19.What are the roots of x² − 12x + 32 = 0? (a) 1 and 32 (b) 2 and 16 (c) 4 and 8 (d) 1 and 11
  20. 20.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3854 (b) 3604 (c) 3844 (d) 3364
  21. 21.What is the sum of the roots of x² − 15x + 56 = 0? (a) 15 (b) 14 (c) 56 (d) 16
  22. 22.If a + b = 10 and ab = 21, what is a² + b²? (a) 142 (b) 58 (c) 100 (d) 79
  23. 23.If a + b = 16 and ab = 55, what is (a − b)²? (a) 256 (b) 36 (c) 146 (d) 6
  24. 24.If a + b = 15 and ab = 56, what is a² + b²? (a) 225 (b) 169 (c) 337 (d) 113
  25. 25.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x − 8)(x − 9) (c) (x + 1)(x + 16) (d) (x + 8)(x + 9)
  26. 26.What is the sum of the roots of x² − 5x + 6 = 0? (a) 6 (b) 4 (c) 7 (d) 5
  27. 27.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x + 1)(x + 13) (c) (x + 6)(x + 8) (d) (x − 6)(x − 8)
  28. 28.What is the sum of the roots of x² − 11x + 18 = 0? (a) 11 (b) 10 (c) 12 (d) 18
  29. 29.What are the roots of x² − 13x + 36 = 0? (a) 2 and 18 (b) 4 and 9 (c) 1 and 36 (d) 1 and 12
  30. 30.If a + b = 5 and ab = 6, what is (a − b)²? (a) 13 (b) 1 (c) 25 (d) 19
  31. 31.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
  32. 32.What is the product of the roots of x² − 9x + 14 = 0? (a) 9 (b) 13 (c) 14 (d) 15
  33. 33.What are the roots of x² − 6x + 8 = 0? (a) 2 and 4 (b) 2 and 5 (c) 1 and 5 (d) 1 and 8
  34. 34.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 6375 (b) 5635 (c) 5625 (d) 7225
  35. 35.What is the product of the roots of x² − 15x + 56 = 0? (a) 57 (b) 15 (c) 55 (d) 56
  36. 36.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 10) (b) (x + 2)(x + 5) (c) (x − 2)(x − 5) (d) (x + 1)(x + 6)
  37. 37.If a + b = 18 and ab = 77, what is (a − b)²? (a) 4 (b) 324 (c) 170 (d) 16
  38. 38.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8104 (b) 8474 (c) 7744 (d) 8464
  39. 39.Which is the correct factorisation of x² + 10x + 24? (a) (x + 4)(x + 6) (b) (x + 1)(x + 24) (c) (x + 1)(x + 9) (d) (x − 4)(x − 6)
  40. 40.What is the product of the roots of x² − 11x + 28 = 0? (a) 11 (b) 27 (c) 29 (d) 28

Answer key

  1. 18
  2. 63
  3. 5 and 6
  4. 16
  5. 16
  6. 784
  7. (x + 4)(x + 7)
  8. 130
  9. (x + 2)(x + 3)
  10. 180
  11. 2704
  12. 13
  13. (x + 2)(x + 9)
  14. 72
  15. 4225
  16. (x + 7)(x + 8)
  17. 54
  18. (x + 2)(x + 8)
  19. 4 and 8
  20. 3844
  21. 15
  22. 58
  23. 36
  24. 113
  25. (x + 8)(x + 9)
  26. 5
  27. (x + 6)(x + 8)
  28. 11
  29. 4 and 9
  30. 1
  31. 6 and 8
  32. 14
  33. 2 and 4
  34. 5625
  35. 56
  36. (x + 2)(x + 5)
  37. 16
  38. 8464
  39. (x + 4)(x + 6)
  40. 28

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