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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 943 · 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 45²? (a) 2035 (b) 2025 (c) 1225 (d) 1625
  2. 2.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 16) (b) (x − 2)(x − 8) (c) (x + 1)(x + 9) (d) (x + 2)(x + 8)
  3. 3.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6724 (b) 6734 (c) 6404 (d) 6084
  4. 4.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3731 (b) 3481 (c) 3601 (d) 3721
  5. 5.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 3575 (b) 4225 (c) 3025 (d) 3035
  6. 6.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 225 (b) 235 (c) 625 (d) 375
  7. 7.If a + b = 15 and ab = 54, what is (a − b)²? (a) 9 (b) 117 (c) 3 (d) 225
  8. 8.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8649 (b) 7569 (c) 8109 (d) 8659
  9. 9.If a + b = 17 and ab = 70, what is a² + b²? (a) 429 (b) 219 (c) 289 (d) 149
  10. 10.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4761 (b) 4771 (c) 4899 (d) 5041
  11. 11.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 1225 (b) 625 (c) 925 (d) 1235
  12. 12.What are the roots of x² − 10x + 21 = 0? (a) 1 and 9 (b) 3 and 7 (c) 2 and 8 (d) 1 and 21
  13. 13.What is the product of the roots of x² − 15x + 56 = 0? (a) 56 (b) 55 (c) 57 (d) 15
  14. 14.If a + b = 14 and ab = 24, what is (a − b)²? (a) 100 (b) 148 (c) 196 (d) 10
  15. 15.If a + b = 5 and ab = 4, what is a² + b²? (a) 25 (b) 33 (c) 17 (d) 21
  16. 16.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 384 (b) 256 (c) 266 (d) 576
  17. 17.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 18) (b) (x + 1)(x + 8) (c) (x − 3)(x − 6) (d) (x + 3)(x + 6)
  18. 18.Which is the correct factorisation of x² + 13x + 42? (a) (x + 6)(x + 7) (b) (x + 1)(x + 12) (c) (x + 1)(x + 42) (d) (x − 6)(x − 7)
  19. 19.What is the sum of the roots of x² − 12x + 32 = 0? (a) 11 (b) 13 (c) 32 (d) 12
  20. 20.If a + b = 10 and ab = 21, what is a² + b²? (a) 142 (b) 79 (c) 100 (d) 58
  21. 21.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 401 (b) 361 (c) 441 (d) 451
  22. 22.If a + b = 10 and ab = 24, what is a² + b²? (a) 100 (b) 76 (c) 52 (d) 148
  23. 23.If a + b = 13 and ab = 40, what is (a − b)²? (a) 169 (b) 89 (c) 3 (d) 9
  24. 24.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 72 (c) 17 (d) 18
  25. 25.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 14) (c) (x + 1)(x + 56) (d) (x + 7)(x + 8)
  26. 26.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 11) (b) (x + 4)(x + 8) (c) (x − 4)(x − 8) (d) (x + 1)(x + 32)
  27. 27.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
  28. 28.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 4925 (b) 5625 (c) 5635 (d) 4225
  29. 29.If a + b = 20 and ab = 96, what is (a − b)²? (a) 16 (b) 400 (c) 208 (d) 4
  30. 30.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 15) (b) (x + 1)(x + 7) (c) (x − 3)(x − 5) (d) (x + 3)(x + 5)
  31. 31.If a + b = 18 and ab = 77, what is (a − b)²? (a) 170 (b) 16 (c) 324 (d) 4
  32. 32.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x + 1)(x + 30) (c) (x − 5)(x − 6) (d) (x + 5)(x + 6)
  33. 33.What are the roots of x² − 15x + 56 = 0? (a) 1 and 56 (b) 2 and 28 (c) 1 and 14 (d) 7 and 8
  34. 34.What is the product of the roots of x² − 11x + 18 = 0? (a) 19 (b) 18 (c) 17 (d) 11
  35. 35.What is the sum of the roots of x² − 10x + 24 = 0? (a) 24 (b) 11 (c) 9 (d) 10
  36. 36.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 5635 (b) 5625 (c) 7225 (d) 6375
  37. 37.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2116 (b) 2516 (c) 2926 (d) 2916
  38. 38.If a + b = 10 and ab = 16, what is a² + b²? (a) 132 (b) 68 (c) 100 (d) 84
  39. 39.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x − 6)(x − 9) (c) (x + 1)(x + 14) (d) (x + 6)(x + 9)
  40. 40.What is the sum of the roots of x² − 15x + 56 = 0? (a) 14 (b) 56 (c) 16 (d) 15

Answer key

  1. 2025
  2. (x + 2)(x + 8)
  3. 6724
  4. 3721
  5. 3025
  6. 225
  7. 9
  8. 8649
  9. 149
  10. 4761
  11. 1225
  12. 3 and 7
  13. 56
  14. 100
  15. 17
  16. 256
  17. (x + 3)(x + 6)
  18. (x + 6)(x + 7)
  19. 12
  20. 58
  21. 441
  22. 52
  23. 9
  24. 17
  25. (x + 7)(x + 8)
  26. (x + 4)(x + 8)
  27. 4 and 7
  28. 5625
  29. 16
  30. (x + 3)(x + 5)
  31. 16
  32. (x + 5)(x + 6)
  33. 7 and 8
  34. 18
  35. 10
  36. 5625
  37. 2916
  38. 68
  39. (x + 6)(x + 9)
  40. 15

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