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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 946 · 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 72²? (a) 5184 (b) 4624 (c) 5194 (d) 4904
  2. 2.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 6375 (b) 5635 (c) 5625 (d) 7225
  3. 3.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x − 7)(x − 9) (c) (x + 1)(x + 15) (d) (x + 7)(x + 9)
  4. 4.If a + b = 8 and ab = 7, what is (a − b)²? (a) 50 (b) 6 (c) 64 (d) 36
  5. 5.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 9801 (b) 10201 (c) 10211 (d) 10001
  6. 6.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
  7. 7.If a + b = 17 and ab = 70, what is (a − b)²? (a) 289 (b) 3 (c) 9 (d) 149
  8. 8.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 7406 (b) 8836 (c) 8084 (d) 7396
  9. 9.If a + b = 12 and ab = 32, what is a² + b²? (a) 208 (b) 144 (c) 80 (d) 112
  10. 10.If a + b = 5 and ab = 4, what is a² + b²? (a) 17 (b) 33 (c) 25 (d) 21
  11. 11.If a + b = 17 and ab = 72, what is (a − b)²? (a) 145 (b) 217 (c) 289 (d) 1
  12. 12.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x + 5)(x + 6) (c) (x + 1)(x + 10) (d) (x − 5)(x − 6)
  13. 13.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 72 (c) 17 (d) 18
  14. 14.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x − 8)(x − 9) (c) (x + 8)(x + 9) (d) (x + 1)(x + 16)
  15. 15.If a + b = 17 and ab = 60, what is a² + b²? (a) 229 (b) 169 (c) 289 (d) 409
  16. 16.If a + b = 15 and ab = 50, what is a² + b²? (a) 225 (b) 175 (c) 125 (d) 325
  17. 17.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)
  18. 18.What is the sum of the roots of x² − 8x + 12 = 0? (a) 9 (b) 12 (c) 8 (d) 7
  19. 19.What are the roots of x² − 9x + 14 = 0? (a) 3 and 6 (b) 1 and 8 (c) 2 and 7 (d) 1 and 14
  20. 20.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1609 (b) 1859 (c) 1369 (d) 1849
  21. 21.What are the roots of x² − 9x + 18 = 0? (a) 1 and 18 (b) 3 and 6 (c) 1 and 8 (d) 2 and 9
  22. 22.If a + b = 7 and ab = 10, what is (a − b)²? (a) 3 (b) 9 (c) 29 (d) 49
  23. 23.Which is the correct factorisation of x² + 14x + 48? (a) (x + 6)(x + 8) (b) (x − 6)(x − 8) (c) (x + 1)(x + 48) (d) (x + 1)(x + 13)
  24. 24.What is the sum of the roots of x² − 9x + 20 = 0? (a) 20 (b) 8 (c) 10 (d) 9
  25. 25.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x + 1)(x + 56) (c) (x − 7)(x − 8) (d) (x + 1)(x + 14)
  26. 26.What are the roots of x² − 10x + 16 = 0? (a) 2 and 8 (b) 1 and 9 (c) 1 and 16 (d) 3 and 7
  27. 27.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 961 (b) 971 (c) 841 (d) 901
  28. 28.What is the sum of the roots of x² − 12x + 27 = 0? (a) 12 (b) 27 (c) 13 (d) 11
  29. 29.If a + b = 11 and ab = 24, what is (a − b)²? (a) 121 (b) 73 (c) 5 (d) 25
  30. 30.If a + b = 8 and ab = 15, what is (a − b)²? (a) 34 (b) 4 (c) 2 (d) 64
  31. 31.What is the product of the roots of x² − 13x + 36 = 0? (a) 13 (b) 37 (c) 35 (d) 36
  32. 32.If a + b = 17 and ab = 66, what is (a − b)²? (a) 289 (b) 157 (c) 5 (d) 25
  33. 33.Which is the correct factorisation of x² + 12x + 32? (a) (x + 4)(x + 8) (b) (x + 1)(x + 32) (c) (x + 1)(x + 11) (d) (x − 4)(x − 8)
  34. 34.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2714 (b) 2704 (c) 2504 (d) 2304
  35. 35.What is the sum of the roots of x² − 10x + 16 = 0? (a) 10 (b) 9 (c) 16 (d) 11
  36. 36.What is the product of the roots of x² − 14x + 48 = 0? (a) 47 (b) 49 (c) 48 (d) 14
  37. 37.If a + b = 11 and ab = 30, what is a² + b²? (a) 91 (b) 181 (c) 121 (d) 61
  38. 38.What is the sum of the roots of x² − 11x + 24 = 0? (a) 12 (b) 24 (c) 10 (d) 11
  39. 39.If a + b = 7 and ab = 12, what is (a − b)²? (a) 25 (b) 49 (c) 37 (d) 1
  40. 40.If a + b = 18 and ab = 72, what is (a − b)²? (a) 324 (b) 180 (c) 6 (d) 36

Answer key

  1. 5184
  2. 5625
  3. (x + 7)(x + 9)
  4. 36
  5. 10201
  6. 7 and 9
  7. 9
  8. 7396
  9. 80
  10. 17
  11. 1
  12. (x + 5)(x + 6)
  13. 17
  14. (x + 8)(x + 9)
  15. 169
  16. 125
  17. (x + 6)(x + 7)
  18. 8
  19. 2 and 7
  20. 1849
  21. 3 and 6
  22. 9
  23. (x + 6)(x + 8)
  24. 9
  25. (x + 7)(x + 8)
  26. 2 and 8
  27. 961
  28. 12
  29. 25
  30. 4
  31. 36
  32. 25
  33. (x + 4)(x + 8)
  34. 2704
  35. 10
  36. 48
  37. 61
  38. 11
  39. 1
  40. 36

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