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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 151 · 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 sum of the roots of x² − 12x + 32 = 0? (a) 32 (b) 13 (c) 11 (d) 12
  2. 2.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 2525 (b) 3025 (c) 3035 (d) 2025
  3. 3.What is the sum of the roots of x² − 7x + 12 = 0? (a) 6 (b) 12 (c) 8 (d) 7
  4. 4.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)
  5. 5.Which is the correct factorisation of x² + 9x + 18? (a) (x + 3)(x + 6) (b) (x + 1)(x + 8) (c) (x + 1)(x + 18) (d) (x − 3)(x − 6)
  6. 6.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 10816 (b) 10826 (c) 9216 (d) 10016
  7. 7.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 971 (b) 901 (c) 961 (d) 841
  8. 8.Which is the correct factorisation of x² + 7x + 10? (a) (x − 2)(x − 5) (b) (x + 1)(x + 10) (c) (x + 1)(x + 6) (d) (x + 2)(x + 5)
  9. 9.If a + b = 13 and ab = 22, what is a² + b²? (a) 147 (b) 213 (c) 125 (d) 169
  10. 10.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 14) (b) (x + 2)(x + 7) (c) (x − 2)(x − 7) (d) (x + 1)(x + 8)
  11. 11.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3596 (b) 3844 (c) 3364 (d) 3374
  12. 12.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10404 (b) 10414 (c) 9604 (d) 10004
  13. 13.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2496 (b) 2304 (c) 2314 (d) 2704
  14. 14.What is the product of the roots of x² − 11x + 24 = 0? (a) 23 (b) 24 (c) 25 (d) 11
  15. 15.If a + b = 19 and ab = 90, what is (a − b)²? (a) 1 (b) 271 (c) 361 (d) 181
  16. 16.What is the product of the roots of x² − 13x + 40 = 0? (a) 40 (b) 13 (c) 39 (d) 41
  17. 17.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x + 1)(x + 10) (c) (x − 5)(x − 6) (d) (x + 5)(x + 6)
  18. 18.If a + b = 11 and ab = 18, what is a² + b²? (a) 103 (b) 121 (c) 85 (d) 157
  19. 19.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 5184 (b) 4634 (c) 4896 (d) 4624
  20. 20.If a + b = 15 and ab = 56, what is (a − b)²? (a) 113 (b) 225 (c) 169 (d) 1
  21. 21.If a + b = 14 and ab = 45, what is (a − b)²? (a) 4 (b) 196 (c) 16 (d) 106
  22. 22.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x − 6)(x − 9) (c) (x + 6)(x + 9) (d) (x + 1)(x + 14)
  23. 23.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 3584 (b) 4096 (c) 3136 (d) 3146
  24. 24.If a + b = 13 and ab = 12, what is a² + b²? (a) 169 (b) 157 (c) 193 (d) 145
  25. 25.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 17 (c) 72 (d) 71
  26. 26.If a + b = 21 and ab = 108, what is a² + b²? (a) 657 (b) 225 (c) 441 (d) 333
  27. 27.What is the sum of the roots of x² − 11x + 18 = 0? (a) 12 (b) 18 (c) 11 (d) 10
  28. 28.If a + b = 12 and ab = 11, what is (a − b)²? (a) 10 (b) 144 (c) 122 (d) 100
  29. 29.If a + b = 15 and ab = 56, what is a² + b²? (a) 225 (b) 113 (c) 169 (d) 337
  30. 30.If a + b = 11 and ab = 30, what is a² + b²? (a) 121 (b) 91 (c) 61 (d) 181
  31. 31.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 909 (b) 1099 (c) 729 (d) 1089
  32. 32.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 361 (b) 401 (c) 451 (d) 441
  33. 33.If a + b = 5 and ab = 4, what is (a − b)²? (a) 17 (b) 3 (c) 25 (d) 9
  34. 34.What is the sum of the roots of x² − 10x + 21 = 0? (a) 11 (b) 21 (c) 9 (d) 10
  35. 35.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6401 (b) 6561 (c) 6241 (d) 6571
  36. 36.What is the sum of the roots of x² − 8x + 15 = 0? (a) 7 (b) 9 (c) 8 (d) 15
  37. 37.Which is the correct factorisation of x² + 14x + 45? (a) (x − 5)(x − 9) (b) (x + 5)(x + 9) (c) (x + 1)(x + 45) (d) (x + 1)(x + 13)
  38. 38.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2116 (b) 2126 (c) 2484 (d) 2916
  39. 39.What is the sum of the roots of x² − 13x + 40 = 0? (a) 12 (b) 13 (c) 40 (d) 14
  40. 40.If a + b = 16 and ab = 63, what is a² + b²? (a) 256 (b) 130 (c) 382 (d) 193

Answer key

  1. 12
  2. 3025
  3. 7
  4. (x + 7)(x + 9)
  5. (x + 3)(x + 6)
  6. 10816
  7. 961
  8. (x + 2)(x + 5)
  9. 125
  10. (x + 2)(x + 7)
  11. 3364
  12. 10404
  13. 2304
  14. 24
  15. 1
  16. 40
  17. (x + 5)(x + 6)
  18. 85
  19. 4624
  20. 1
  21. 16
  22. (x + 6)(x + 9)
  23. 3136
  24. 145
  25. 72
  26. 225
  27. 11
  28. 100
  29. 113
  30. 61
  31. 1089
  32. 441
  33. 9
  34. 10
  35. 6561
  36. 8
  37. (x + 5)(x + 9)
  38. 2116
  39. 13
  40. 130

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