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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 723 · 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 = 30, what is (a − b)²? (a) 169 (b) 7 (c) 109 (d) 49
  2. 2.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 72) (c) (x + 8)(x + 9) (d) (x + 1)(x + 16)
  3. 3.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)
  4. 4.What is the product of the roots of x² − 11x + 30 = 0? (a) 31 (b) 30 (c) 11 (d) 29
  5. 5.If a + b = 13 and ab = 22, what is (a − b)²? (a) 81 (b) 169 (c) 9 (d) 125
  6. 6.Which is the correct factorisation of x² + 12x + 35? (a) (x − 5)(x − 7) (b) (x + 1)(x + 35) (c) (x + 1)(x + 11) (d) (x + 5)(x + 7)
  7. 7.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 24) (b) (x + 1)(x + 9) (c) (x − 4)(x − 6) (d) (x + 4)(x + 6)
  8. 8.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10404 (b) 10414 (c) 10004 (d) 9604
  9. 9.Which is the correct factorisation of x² + 11x + 18? (a) (x + 2)(x + 9) (b) (x + 1)(x + 10) (c) (x + 1)(x + 18) (d) (x − 2)(x − 9)
  10. 10.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 6) (b) (x + 2)(x + 5) (c) (x + 1)(x + 10) (d) (x − 2)(x − 5)
  11. 11.Which is the correct factorisation of x² + 10x + 16? (a) (x + 2)(x + 8) (b) (x − 2)(x − 8) (c) (x + 1)(x + 16) (d) (x + 1)(x + 9)
  12. 12.If a + b = 10 and ab = 24, what is (a − b)²? (a) 4 (b) 2 (c) 52 (d) 100
  13. 13.If a + b = 9 and ab = 8, what is a² + b²? (a) 81 (b) 97 (c) 65 (d) 73
  14. 14.If a + b = 7 and ab = 12, what is (a − b)²? (a) 37 (b) 25 (c) 49 (d) 1
  15. 15.If a + b = 18 and ab = 72, what is (a − b)²? (a) 180 (b) 324 (c) 6 (d) 36
  16. 16.What is the product of the roots of x² − 15x + 56 = 0? (a) 15 (b) 57 (c) 55 (d) 56
  17. 17.What is the product of the roots of x² − 5x + 6 = 0? (a) 8 (b) 6 (c) 7 (d) 5
  18. 18.If a + b = 11 and ab = 30, what is a² + b²? (a) 121 (b) 61 (c) 91 (d) 181
  19. 19.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9975 (b) 11025 (c) 9035 (d) 9025
  20. 20.If a + b = 16 and ab = 55, what is (a − b)²? (a) 256 (b) 6 (c) 36 (d) 146
  21. 21.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 18 (c) 17 (d) 72
  22. 22.If a + b = 13 and ab = 36, what is (a − b)²? (a) 5 (b) 169 (c) 97 (d) 25
  23. 23.What is the product of the roots of x² − 13x + 42 = 0? (a) 13 (b) 41 (c) 42 (d) 43
  24. 24.If a + b = 13 and ab = 42, what is (a − b)²? (a) 1 (b) 169 (c) 85 (d) 127
  25. 25.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 334 (b) 324 (c) 396 (d) 484
  26. 26.If a + b = 15 and ab = 44, what is a² + b²? (a) 225 (b) 137 (c) 313 (d) 181
  27. 27.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x − 4)(x − 5) (c) (x + 1)(x + 8) (d) (x + 4)(x + 5)
  28. 28.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 73 (c) 72 (d) 17
  29. 29.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8101 (b) 8291 (c) 7921 (d) 8281
  30. 30.Which is the correct factorisation of x² + 11x + 28? (a) (x − 4)(x − 7) (b) (x + 1)(x + 10) (c) (x + 1)(x + 28) (d) (x + 4)(x + 7)
  31. 31.What are the roots of x² − 15x + 56 = 0? (a) 2 and 28 (b) 7 and 8 (c) 1 and 14 (d) 1 and 56
  32. 32.If a + b = 19 and ab = 90, what is a² + b²? (a) 361 (b) 271 (c) 541 (d) 181
  33. 33.If a + b = 19 and ab = 84, what is a² + b²? (a) 529 (b) 361 (c) 277 (d) 193
  34. 34.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6724 (b) 6094 (c) 6396 (d) 6084
  35. 35.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4771 (b) 5041 (c) 4761 (d) 4899
  36. 36.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 7235 (b) 8075 (c) 9025 (d) 7225
  37. 37.What is the product of the roots of x² − 10x + 24 = 0? (a) 10 (b) 23 (c) 25 (d) 24
  38. 38.If a + b = 20 and ab = 96, what is (a − b)²? (a) 400 (b) 208 (c) 4 (d) 16
  39. 39.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 10816 (b) 9216 (c) 10016 (d) 10826
  40. 40.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 8281 (b) 7931 (c) 7921 (d) 8099

Answer key

  1. 49
  2. (x + 8)(x + 9)
  3. (x + 7)(x + 8)
  4. 30
  5. 81
  6. (x + 5)(x + 7)
  7. (x + 4)(x + 6)
  8. 10404
  9. (x + 2)(x + 9)
  10. (x + 2)(x + 5)
  11. (x + 2)(x + 8)
  12. 4
  13. 65
  14. 1
  15. 36
  16. 56
  17. 6
  18. 61
  19. 9025
  20. 36
  21. 17
  22. 25
  23. 42
  24. 1
  25. 324
  26. 137
  27. (x + 4)(x + 5)
  28. 72
  29. 8281
  30. (x + 4)(x + 7)
  31. 7 and 8
  32. 181
  33. 193
  34. 6084
  35. 4761
  36. 7225
  37. 24
  38. 16
  39. 10816
  40. 7921

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