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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 562 · 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.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x + 6)(x + 8) (c) (x + 1)(x + 13) (d) (x − 6)(x − 8)
  2. 2.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 56) (c) (x + 1)(x + 14) (d) (x + 7)(x + 8)
  3. 3.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 4916 (b) 5486 (c) 4356 (d) 5476
  4. 4.If a + b = 10 and ab = 16, what is (a − b)²? (a) 36 (b) 68 (c) 100 (d) 6
  5. 5.If a + b = 20 and ab = 96, what is (a − b)²? (a) 208 (b) 400 (c) 4 (d) 16
  6. 6.If a + b = 6 and ab = 8, what is a² + b²? (a) 52 (b) 28 (c) 36 (d) 20
  7. 7.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 4909 (b) 5329 (c) 5339 (d) 4489
  8. 8.What are the roots of x² − 14x + 48 = 0? (a) 6 and 8 (b) 1 and 48 (c) 2 and 24 (d) 1 and 13
  9. 9.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 1166 (b) 676 (c) 1156 (d) 916
  10. 10.What is the sum of the roots of x² − 10x + 16 = 0? (a) 11 (b) 9 (c) 16 (d) 10
  11. 11.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 2035 (b) 1625 (c) 1225 (d) 2025
  12. 12.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8101 (b) 8291 (c) 8281 (d) 7921
  13. 13.If a + b = 7 and ab = 6, what is (a − b)²? (a) 25 (b) 37 (c) 49 (d) 5
  14. 14.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 4235 (b) 3625 (c) 3025 (d) 4225
  15. 15.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x + 1)(x + 42) (c) (x + 6)(x + 7) (d) (x − 6)(x − 7)
  16. 16.If a + b = 14 and ab = 24, what is (a − b)²? (a) 10 (b) 196 (c) 100 (d) 148
  17. 17.What is the product of the roots of x² − 11x + 28 = 0? (a) 29 (b) 28 (c) 27 (d) 11
  18. 18.If a + b = 7 and ab = 10, what is a² + b²? (a) 29 (b) 69 (c) 39 (d) 49
  19. 19.What is the product of the roots of x² − 14x + 48 = 0? (a) 47 (b) 49 (c) 14 (d) 48
  20. 20.If a + b = 14 and ab = 33, what is (a − b)²? (a) 64 (b) 196 (c) 130 (d) 8
  21. 21.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x − 8)(x − 9) (c) (x + 1)(x + 16) (d) (x + 8)(x + 9)
  22. 22.Which is the correct factorisation of x² + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 5)(x + 6) (c) (x + 1)(x + 10) (d) (x + 1)(x + 30)
  23. 23.What is the product of the roots of x² − 15x + 54 = 0? (a) 15 (b) 55 (c) 54 (d) 53
  24. 24.If a + b = 16 and ab = 55, what is (a − b)²? (a) 6 (b) 256 (c) 36 (d) 146
  25. 25.What is the sum of the roots of x² − 15x + 54 = 0? (a) 16 (b) 15 (c) 14 (d) 54
  26. 26.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3979 (b) 3249 (c) 3609 (d) 3969
  27. 27.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 7) (b) (x + 1)(x + 12) (c) (x − 2)(x − 6) (d) (x + 2)(x + 6)
  28. 28.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)
  29. 29.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2126 (b) 2916 (c) 2116 (d) 2484
  30. 30.If a + b = 17 and ab = 60, what is a² + b²? (a) 169 (b) 289 (c) 229 (d) 409
  31. 31.If a + b = 8 and ab = 15, what is a² + b²? (a) 34 (b) 94 (c) 64 (d) 49
  32. 32.If a + b = 13 and ab = 22, what is (a − b)²? (a) 81 (b) 169 (c) 9 (d) 125
  33. 33.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3374 (b) 3844 (c) 3364 (d) 3596
  34. 34.If a + b = 19 and ab = 90, what is (a − b)²? (a) 271 (b) 181 (c) 361 (d) 1
  35. 35.If a + b = 14 and ab = 48, what is (a − b)²? (a) 2 (b) 4 (c) 100 (d) 196
  36. 36.If a + b = 11 and ab = 28, what is a² + b²? (a) 121 (b) 93 (c) 65 (d) 177
  37. 37.What is the sum of the roots of x² − 8x + 12 = 0? (a) 12 (b) 7 (c) 9 (d) 8
  38. 38.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9604 (b) 10404 (c) 9614 (d) 9996
  39. 39.If a + b = 10 and ab = 21, what is (a − b)²? (a) 58 (b) 4 (c) 16 (d) 100
  40. 40.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6401 (b) 6571 (c) 6241 (d) 6561

Answer key

  1. (x + 6)(x + 8)
  2. (x + 7)(x + 8)
  3. 5476
  4. 36
  5. 16
  6. 20
  7. 5329
  8. 6 and 8
  9. 1156
  10. 10
  11. 2025
  12. 8281
  13. 25
  14. 4225
  15. (x + 6)(x + 7)
  16. 100
  17. 28
  18. 29
  19. 48
  20. 64
  21. (x + 8)(x + 9)
  22. (x + 5)(x + 6)
  23. 54
  24. 36
  25. 15
  26. 3969
  27. (x + 2)(x + 6)
  28. (x + 4)(x + 6)
  29. 2116
  30. 169
  31. 34
  32. 81
  33. 3364
  34. 1
  35. 4
  36. 65
  37. 8
  38. 9604
  39. 16
  40. 6561

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