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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 99 · 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 82²? (a) 6084 (b) 6724 (c) 6734 (d) 6404
  2. 2.What are the roots of x² − 12x + 32 = 0? (a) 1 and 11 (b) 2 and 16 (c) 1 and 32 (d) 4 and 8
  3. 3.If a + b = 11 and ab = 28, what is (a − b)²? (a) 65 (b) 121 (c) 3 (d) 9
  4. 4.Which is the correct factorisation of x² + 6x + 8? (a) (x + 2)(x + 4) (b) (x − 2)(x − 4) (c) (x + 1)(x + 8) (d) (x + 1)(x + 5)
  5. 5.What are the roots of x² − 16x + 63 = 0? (a) 3 and 21 (b) 1 and 15 (c) 7 and 9 (d) 1 and 63
  6. 6.What are the roots of x² − 9x + 20 = 0? (a) 1 and 8 (b) 2 and 10 (c) 1 and 20 (d) 4 and 5
  7. 7.If a + b = 12 and ab = 27, what is a² + b²? (a) 198 (b) 144 (c) 90 (d) 117
  8. 8.Which is the correct factorisation of x² + 11x + 28? (a) (x + 4)(x + 7) (b) (x − 4)(x − 7) (c) (x + 1)(x + 28) (d) (x + 1)(x + 10)
  9. 9.What are the roots of x² − 5x + 6 = 0? (a) 1 and 6 (b) 1 and 4 (c) 2 and 4 (d) 2 and 3
  10. 10.What are the roots of x² − 10x + 16 = 0? (a) 2 and 8 (b) 1 and 16 (c) 3 and 7 (d) 1 and 9
  11. 11.If a + b = 11 and ab = 30, what is a² + b²? (a) 61 (b) 181 (c) 121 (d) 91
  12. 12.Which is the correct factorisation of x² + 12x + 32? (a) (x − 4)(x − 8) (b) (x + 1)(x + 32) (c) (x + 4)(x + 8) (d) (x + 1)(x + 11)
  13. 13.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 72 (c) 17 (d) 16
  14. 14.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 2025 (b) 2525 (c) 3025 (d) 3035
  15. 15.If a + b = 12 and ab = 20, what is (a − b)²? (a) 8 (b) 144 (c) 64 (d) 104
  16. 16.What is the product of the roots of x² − 15x + 56 = 0? (a) 15 (b) 55 (c) 56 (d) 57
  17. 17.If a + b = 13 and ab = 30, what is a² + b²? (a) 109 (b) 169 (c) 139 (d) 229
  18. 18.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 8281 (b) 7931 (c) 8099 (d) 7921
  19. 19.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 371 (b) 441 (c) 361 (d) 399
  20. 20.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 8836 (b) 8084 (c) 7396 (d) 7406
  21. 21.What are the roots of x² − 14x + 48 = 0? (a) 1 and 13 (b) 2 and 24 (c) 6 and 8 (d) 1 and 48
  22. 22.If a + b = 3 and ab = 2, what is (a − b)²? (a) 1 (b) 5 (c) 9 (d) 7
  23. 23.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)
  24. 24.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 2 and 36 (c) 8 and 9 (d) 1 and 16
  25. 25.What is the sum of the roots of x² − 12x + 27 = 0? (a) 11 (b) 27 (c) 13 (d) 12
  26. 26.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3969 (b) 3979 (c) 3249 (d) 3609
  27. 27.If a + b = 11 and ab = 24, what is a² + b²? (a) 97 (b) 169 (c) 73 (d) 121
  28. 28.What is the sum of the roots of x² − 15x + 56 = 0? (a) 15 (b) 16 (c) 56 (d) 14
  29. 29.If a + b = 13 and ab = 12, what is (a − b)²? (a) 169 (b) 121 (c) 145 (d) 11
  30. 30.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x + 1)(x + 15) (c) (x + 7)(x + 9) (d) (x − 7)(x − 9)
  31. 31.What is the sum of the roots of x² − 14x + 48 = 0? (a) 48 (b) 15 (c) 13 (d) 14
  32. 32.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 73 (c) 17 (d) 72
  33. 33.If a + b = 19 and ab = 90, what is a² + b²? (a) 541 (b) 181 (c) 271 (d) 361
  34. 34.What is the product of the roots of x² − 10x + 16 = 0? (a) 10 (b) 15 (c) 16 (d) 17
  35. 35.If a + b = 6 and ab = 8, what is (a − b)²? (a) 20 (b) 36 (c) 2 (d) 4
  36. 36.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 1)(x + 12) (c) (x + 6)(x + 7) (d) (x + 1)(x + 42)
  37. 37.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 5051 (b) 4901 (c) 4761 (d) 5041
  38. 38.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 9) (b) (x − 4)(x − 6) (c) (x + 1)(x + 24) (d) (x + 4)(x + 6)
  39. 39.If a + b = 14 and ab = 24, what is a² + b²? (a) 172 (b) 196 (c) 148 (d) 244
  40. 40.If a + b = 8 and ab = 15, what is a² + b²? (a) 64 (b) 49 (c) 94 (d) 34

Answer key

  1. 6724
  2. 4 and 8
  3. 9
  4. (x + 2)(x + 4)
  5. 7 and 9
  6. 4 and 5
  7. 90
  8. (x + 4)(x + 7)
  9. 2 and 3
  10. 2 and 8
  11. 61
  12. (x + 4)(x + 8)
  13. 17
  14. 3025
  15. 64
  16. 56
  17. 109
  18. 7921
  19. 361
  20. 7396
  21. 6 and 8
  22. 1
  23. (x + 8)(x + 9)
  24. 8 and 9
  25. 12
  26. 3969
  27. 73
  28. 15
  29. 121
  30. (x + 7)(x + 9)
  31. 14
  32. 72
  33. 181
  34. 16
  35. 4
  36. (x + 6)(x + 7)
  37. 5041
  38. (x + 4)(x + 6)
  39. 148
  40. 34

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