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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 197 · 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 product of the roots of x² − 12x + 35 = 0? (a) 34 (b) 35 (c) 36 (d) 12
  2. 2.What is the sum of the roots of x² − 11x + 24 = 0? (a) 12 (b) 24 (c) 10 (d) 11
  3. 3.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6084 (b) 6404 (c) 6734 (d) 6724
  4. 4.If a + b = 5 and ab = 4, what is (a − b)²? (a) 25 (b) 17 (c) 3 (d) 9
  5. 5.What is the sum of the roots of x² − 6x + 8 = 0? (a) 8 (b) 6 (c) 5 (d) 7
  6. 6.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 586 (b) 256 (c) 416 (d) 576
  7. 7.If a + b = 8 and ab = 12, what is a² + b²? (a) 64 (b) 88 (c) 52 (d) 40
  8. 8.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2611 (b) 2401 (c) 2501 (d) 2601
  9. 9.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 841 (b) 961 (c) 901 (d) 971
  10. 10.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 14) (b) (x + 1)(x + 8) (c) (x + 2)(x + 7) (d) (x − 2)(x − 7)
  11. 11.If a + b = 4 and ab = 3, what is a² + b²? (a) 13 (b) 16 (c) 22 (d) 10
  12. 12.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)
  13. 13.What is the sum of the roots of x² − 11x + 18 = 0? (a) 10 (b) 11 (c) 12 (d) 18
  14. 14.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 16) (c) (x + 1)(x + 72) (d) (x + 8)(x + 9)
  15. 15.If a + b = 10 and ab = 16, what is (a − b)²? (a) 6 (b) 68 (c) 36 (d) 100
  16. 16.If a + b = 13 and ab = 40, what is a² + b²? (a) 249 (b) 129 (c) 169 (d) 89
  17. 17.If a + b = 8 and ab = 7, what is a² + b²? (a) 78 (b) 64 (c) 50 (d) 57
  18. 18.Which is the correct factorisation of x² + 12x + 35? (a) (x − 5)(x − 7) (b) (x + 5)(x + 7) (c) (x + 1)(x + 11) (d) (x + 1)(x + 35)
  19. 19.What is the sum of the roots of x² − 11x + 30 = 0? (a) 11 (b) 10 (c) 30 (d) 12
  20. 20.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 7744 (b) 8104 (c) 8474 (d) 8464
  21. 21.If a + b = 19 and ab = 90, what is a² + b²? (a) 541 (b) 361 (c) 181 (d) 271
  22. 22.If a + b = 16 and ab = 63, what is (a − b)²? (a) 4 (b) 256 (c) 2 (d) 130
  23. 23.What is the sum of the roots of x² − 12x + 32 = 0? (a) 12 (b) 11 (c) 13 (d) 32
  24. 24.If a + b = 11 and ab = 24, what is (a − b)²? (a) 73 (b) 121 (c) 25 (d) 5
  25. 25.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6724 (b) 6396 (c) 6094 (d) 6084
  26. 26.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 14) (b) (x + 6)(x + 9) (c) (x − 6)(x − 9) (d) (x + 1)(x + 54)
  27. 27.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 24) (b) (x + 1)(x + 10) (c) (x + 3)(x + 8) (d) (x − 3)(x − 8)
  28. 28.What is the sum of the roots of x² − 14x + 48 = 0? (a) 15 (b) 14 (c) 13 (d) 48
  29. 29.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 676 (b) 1156 (c) 916 (d) 1166
  30. 30.If a + b = 18 and ab = 77, what is a² + b²? (a) 170 (b) 324 (c) 478 (d) 247
  31. 31.If a + b = 20 and ab = 96, what is a² + b²? (a) 400 (b) 304 (c) 208 (d) 592
  32. 32.What is the product of the roots of x² − 15x + 56 = 0? (a) 55 (b) 57 (c) 56 (d) 15
  33. 33.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 1)(x + 42) (c) (x + 1)(x + 12) (d) (x + 6)(x + 7)
  34. 34.If a + b = 21 and ab = 108, what is (a − b)²? (a) 441 (b) 3 (c) 225 (d) 9
  35. 35.Which is the correct factorisation of x² + 12x + 32? (a) (x + 4)(x + 8) (b) (x − 4)(x − 8) (c) (x + 1)(x + 11) (d) (x + 1)(x + 32)
  36. 36.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 2 and 36 (c) 1 and 72 (d) 8 and 9
  37. 37.If a + b = 15 and ab = 56, what is a² + b²? (a) 337 (b) 225 (c) 113 (d) 169
  38. 38.Which is the correct factorisation of x² + 10x + 24? (a) (x − 4)(x − 6) (b) (x + 1)(x + 24) (c) (x + 1)(x + 9) (d) (x + 4)(x + 6)
  39. 39.Which is the correct factorisation of x² + 10x + 16? (a) (x − 2)(x − 8) (b) (x + 1)(x + 9) (c) (x + 1)(x + 16) (d) (x + 2)(x + 8)
  40. 40.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 8836 (b) 7396 (c) 8846 (d) 8116

Answer key

  1. 35
  2. 11
  3. 6724
  4. 9
  5. 6
  6. 576
  7. 40
  8. 2601
  9. 961
  10. (x + 2)(x + 7)
  11. 10
  12. (x + 7)(x + 9)
  13. 11
  14. (x + 8)(x + 9)
  15. 36
  16. 89
  17. 50
  18. (x + 5)(x + 7)
  19. 11
  20. 8464
  21. 181
  22. 4
  23. 12
  24. 25
  25. 6084
  26. (x + 6)(x + 9)
  27. (x + 3)(x + 8)
  28. 14
  29. 1156
  30. 170
  31. 208
  32. 56
  33. (x + 6)(x + 7)
  34. 9
  35. (x + 4)(x + 8)
  36. 8 and 9
  37. 113
  38. (x + 4)(x + 6)
  39. (x + 2)(x + 8)
  40. 8836

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