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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 808 · 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 = 13 and ab = 12, what is (a − b)²? (a) 11 (b) 145 (c) 121 (d) 169
  2. 2.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 676 (b) 1156 (c) 1166 (d) 916
  3. 3.What is the sum of the roots of x² − 10x + 24 = 0? (a) 9 (b) 11 (c) 24 (d) 10
  4. 4.What is the sum of the roots of x² − 11x + 28 = 0? (a) 11 (b) 10 (c) 12 (d) 28
  5. 5.What is the product of the roots of x² − 5x + 6 = 0? (a) 7 (b) 6 (c) 5 (d) 8
  6. 6.If a + b = 7 and ab = 6, what is (a − b)²? (a) 49 (b) 37 (c) 25 (d) 5
  7. 7.What are the roots of x² − 11x + 18 = 0? (a) 1 and 18 (b) 3 and 6 (c) 1 and 10 (d) 2 and 9
  8. 8.If a + b = 14 and ab = 24, what is a² + b²? (a) 172 (b) 196 (c) 244 (d) 148
  9. 9.If a + b = 20 and ab = 96, what is a² + b²? (a) 592 (b) 208 (c) 304 (d) 400
  10. 10.If a + b = 20 and ab = 96, what is (a − b)²? (a) 4 (b) 400 (c) 16 (d) 208
  11. 11.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 441 (b) 399 (c) 361 (d) 371
  12. 12.What is the product of the roots of x² − 9x + 18 = 0? (a) 9 (b) 17 (c) 19 (d) 18
  13. 13.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 8649 (b) 7579 (c) 8091 (d) 7569
  14. 14.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x + 6)(x + 8) (c) (x + 1)(x + 48) (d) (x − 6)(x − 8)
  15. 15.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 971 (b) 901 (c) 961 (d) 841
  16. 16.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10009 (b) 10609 (c) 10619 (d) 9409
  17. 17.What is the sum of the roots of x² − 14x + 45 = 0? (a) 45 (b) 13 (c) 14 (d) 15
  18. 18.If a + b = 13 and ab = 42, what is a² + b²? (a) 85 (b) 169 (c) 127 (d) 253
  19. 19.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 73 (c) 17 (d) 72
  20. 20.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2516 (b) 2116 (c) 2926 (d) 2916
  21. 21.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x − 7)(x − 9) (c) (x + 1)(x + 63) (d) (x + 1)(x + 15)
  22. 22.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 12) (b) (x + 1)(x + 7) (c) (x − 2)(x − 6) (d) (x + 2)(x + 6)
  23. 23.If a + b = 12 and ab = 20, what is (a − b)²? (a) 64 (b) 8 (c) 104 (d) 144
  24. 24.Which is the correct factorisation of x² + 9x + 18? (a) (x − 3)(x − 6) (b) (x + 3)(x + 6) (c) (x + 1)(x + 18) (d) (x + 1)(x + 8)
  25. 25.If a + b = 13 and ab = 30, what is a² + b²? (a) 229 (b) 169 (c) 109 (d) 139
  26. 26.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 3025 (b) 4235 (c) 3625 (d) 4225
  27. 27.If a + b = 3 and ab = 2, what is a² + b²? (a) 13 (b) 5 (c) 7 (d) 9
  28. 28.If a + b = 18 and ab = 77, what is (a − b)²? (a) 16 (b) 324 (c) 4 (d) 170
  29. 29.If a + b = 16 and ab = 55, what is (a − b)²? (a) 256 (b) 6 (c) 36 (d) 146
  30. 30.What is the product of the roots of x² − 14x + 48 = 0? (a) 14 (b) 49 (c) 48 (d) 47
  31. 31.What is the product of the roots of x² − 11x + 30 = 0? (a) 31 (b) 29 (c) 11 (d) 30
  32. 32.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 7)(x + 8) (c) (x + 1)(x + 14) (d) (x − 7)(x − 8)
  33. 33.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x − 6)(x − 7) (c) (x + 1)(x + 42) (d) (x + 6)(x + 7)
  34. 34.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 10016 (b) 9216 (c) 10826 (d) 10816
  35. 35.What are the roots of x² − 16x + 63 = 0? (a) 1 and 63 (b) 1 and 15 (c) 3 and 21 (d) 7 and 9
  36. 36.What is the sum of the roots of x² − 15x + 56 = 0? (a) 15 (b) 14 (c) 56 (d) 16
  37. 37.What are the roots of x² − 7x + 12 = 0? (a) 2 and 6 (b) 1 and 12 (c) 1 and 6 (d) 3 and 4
  38. 38.If a + b = 13 and ab = 42, what is (a − b)²? (a) 127 (b) 169 (c) 85 (d) 1
  39. 39.What is the product of the roots of x² − 8x + 12 = 0? (a) 8 (b) 11 (c) 12 (d) 13
  40. 40.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 7744 (b) 8104 (c) 8474 (d) 8464

Answer key

  1. 121
  2. 1156
  3. 10
  4. 11
  5. 6
  6. 25
  7. 2 and 9
  8. 148
  9. 208
  10. 16
  11. 361
  12. 18
  13. 7569
  14. (x + 6)(x + 8)
  15. 961
  16. 10609
  17. 14
  18. 85
  19. 72
  20. 2916
  21. (x + 7)(x + 9)
  22. (x + 2)(x + 6)
  23. 64
  24. (x + 3)(x + 6)
  25. 109
  26. 4225
  27. 5
  28. 16
  29. 36
  30. 48
  31. 30
  32. (x + 7)(x + 8)
  33. (x + 6)(x + 7)
  34. 10816
  35. 7 and 9
  36. 15
  37. 3 and 4
  38. 1
  39. 12
  40. 8464

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