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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 711 · 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 = 11 and ab = 24, what is (a − b)²? (a) 121 (b) 5 (c) 73 (d) 25
  2. 2.What is the sum of the roots of x² − 7x + 10 = 0? (a) 8 (b) 10 (c) 6 (d) 7
  3. 3.What is the product of the roots of x² − 6x + 8 = 0? (a) 8 (b) 6 (c) 7 (d) 9
  4. 4.If a + b = 13 and ab = 40, what is a² + b²? (a) 169 (b) 249 (c) 89 (d) 129
  5. 5.What is the sum of the roots of x² − 12x + 35 = 0? (a) 11 (b) 13 (c) 12 (d) 35
  6. 6.If a + b = 5 and ab = 4, what is (a − b)²? (a) 17 (b) 9 (c) 3 (d) 25
  7. 7.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 875 (b) 1225 (c) 635 (d) 625
  8. 8.Which is the correct factorisation of x² + 15x + 54? (a) (x + 6)(x + 9) (b) (x − 6)(x − 9) (c) (x + 1)(x + 14) (d) (x + 1)(x + 54)
  9. 9.What is the sum of the roots of x² − 5x + 6 = 0? (a) 7 (b) 4 (c) 6 (d) 5
  10. 10.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 8) (b) (x + 1)(x + 5) (c) (x − 2)(x − 4) (d) (x + 2)(x + 4)
  11. 11.If a + b = 7 and ab = 10, what is a² + b²? (a) 69 (b) 39 (c) 29 (d) 49
  12. 12.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x + 6)(x + 7) (c) (x + 1)(x + 12) (d) (x − 6)(x − 7)
  13. 13.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 7744 (b) 7754 (c) 8464 (d) 8096
  14. 14.If a + b = 19 and ab = 90, what is (a − b)²? (a) 271 (b) 1 (c) 181 (d) 361
  15. 15.If a + b = 9 and ab = 8, what is (a − b)²? (a) 7 (b) 81 (c) 65 (d) 49
  16. 16.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 901 (b) 971 (c) 841 (d) 961
  17. 17.What is the sum of the roots of x² − 13x + 42 = 0? (a) 42 (b) 13 (c) 12 (d) 14
  18. 18.What is the product of the roots of x² − 7x + 12 = 0? (a) 13 (b) 11 (c) 7 (d) 12
  19. 19.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 7)(x + 8) (c) (x + 1)(x + 14) (d) (x + 1)(x + 56)
  20. 20.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x − 8)(x − 9) (c) (x + 1)(x + 16) (d) (x + 1)(x + 72)
  21. 21.What are the roots of x² − 11x + 28 = 0? (a) 1 and 10 (b) 4 and 7 (c) 2 and 14 (d) 1 and 28
  22. 22.If a + b = 19 and ab = 90, what is a² + b²? (a) 271 (b) 361 (c) 181 (d) 541
  23. 23.What are the roots of x² − 14x + 45 = 0? (a) 1 and 45 (b) 3 and 15 (c) 5 and 9 (d) 1 and 13
  24. 24.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 16 (c) 15 (d) 14
  25. 25.If a + b = 11 and ab = 28, what is a² + b²? (a) 177 (b) 65 (c) 93 (d) 121
  26. 26.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 2025 (b) 3035 (c) 2525 (d) 3025
  27. 27.If a + b = 7 and ab = 6, what is (a − b)²? (a) 37 (b) 49 (c) 5 (d) 25
  28. 28.If a + b = 19 and ab = 84, what is (a − b)²? (a) 193 (b) 5 (c) 361 (d) 25
  29. 29.What is the product of the roots of x² − 10x + 24 = 0? (a) 23 (b) 25 (c) 24 (d) 10
  30. 30.What is the sum of the roots of x² − 13x + 36 = 0? (a) 14 (b) 13 (c) 12 (d) 36
  31. 31.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 17 (c) 71 (d) 72
  32. 32.Which is the correct factorisation of x² + 10x + 24? (a) (x + 4)(x + 6) (b) (x + 1)(x + 9) (c) (x + 1)(x + 24) (d) (x − 4)(x − 6)
  33. 33.If a + b = 7 and ab = 10, what is (a − b)²? (a) 3 (b) 9 (c) 49 (d) 29
  34. 34.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3599 (b) 3481 (c) 3721 (d) 3491
  35. 35.Which is the correct factorisation of x² + 5x + 6? (a) (x − 2)(x − 3) (b) (x + 1)(x + 6) (c) (x + 1)(x + 4) (d) (x + 2)(x + 3)
  36. 36.What is the product of the roots of x² − 15x + 54 = 0? (a) 55 (b) 54 (c) 15 (d) 53
  37. 37.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 451 (b) 441 (c) 361 (d) 401
  38. 38.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 5625 (b) 7235 (c) 7225 (d) 6425
  39. 39.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 904 (b) 784 (c) 1034 (d) 1024
  40. 40.If a + b = 19 and ab = 88, what is (a − b)²? (a) 3 (b) 9 (c) 361 (d) 185

Answer key

  1. 25
  2. 7
  3. 8
  4. 89
  5. 12
  6. 9
  7. 625
  8. (x + 6)(x + 9)
  9. 5
  10. (x + 2)(x + 4)
  11. 29
  12. (x + 6)(x + 7)
  13. 7744
  14. 1
  15. 49
  16. 961
  17. 13
  18. 12
  19. (x + 7)(x + 8)
  20. (x + 8)(x + 9)
  21. 4 and 7
  22. 181
  23. 5 and 9
  24. 15
  25. 65
  26. 3025
  27. 25
  28. 25
  29. 24
  30. 13
  31. 72
  32. (x + 4)(x + 6)
  33. 9
  34. 3481
  35. (x + 2)(x + 3)
  36. 54
  37. 441
  38. 7225
  39. 1024
  40. 9

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