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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 837 · 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 + 13x + 42? (a) (x + 1)(x + 42) (b) (x + 1)(x + 12) (c) (x + 6)(x + 7) (d) (x − 6)(x − 7)
  2. 2.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10001 (b) 10211 (c) 9801 (d) 10201
  3. 3.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 1024 (b) 904 (c) 784 (d) 1034
  4. 4.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 12) (b) (x − 2)(x − 6) (c) (x + 1)(x + 7) (d) (x + 2)(x + 6)
  5. 5.If a + b = 13 and ab = 22, what is a² + b²? (a) 169 (b) 213 (c) 147 (d) 125
  6. 6.What is the sum of the roots of x² − 16x + 63 = 0? (a) 17 (b) 15 (c) 16 (d) 63
  7. 7.If a + b = 11 and ab = 30, what is (a − b)²? (a) 61 (b) 1 (c) 121 (d) 91
  8. 8.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 225 (b) 625 (c) 425 (d) 635
  9. 9.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6404 (b) 6084 (c) 6724 (d) 6734
  10. 10.If a + b = 15 and ab = 54, what is a² + b²? (a) 225 (b) 171 (c) 117 (d) 333
  11. 11.What are the roots of x² − 13x + 42 = 0? (a) 6 and 7 (b) 1 and 12 (c) 1 and 42 (d) 2 and 21
  12. 12.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9614 (b) 10404 (c) 9604 (d) 9996
  13. 13.What is the product of the roots of x² − 15x + 56 = 0? (a) 57 (b) 56 (c) 55 (d) 15
  14. 14.Which is the correct factorisation of x² + 12x + 27? (a) (x + 3)(x + 9) (b) (x + 1)(x + 11) (c) (x − 3)(x − 9) (d) (x + 1)(x + 27)
  15. 15.If a + b = 5 and ab = 4, what is (a − b)²? (a) 9 (b) 17 (c) 3 (d) 25
  16. 16.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 925 (b) 625 (c) 1235 (d) 1225
  17. 17.What is the sum of the roots of x² − 8x + 12 = 0? (a) 7 (b) 12 (c) 8 (d) 9
  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 − 10x + 21 = 0? (a) 11 (b) 9 (c) 10 (d) 21
  20. 20.What are the roots of x² − 13x + 36 = 0? (a) 2 and 18 (b) 1 and 36 (c) 4 and 9 (d) 1 and 12
  21. 21.If a + b = 10 and ab = 9, what is a² + b²? (a) 100 (b) 118 (c) 82 (d) 91
  22. 22.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2601 (b) 2401 (c) 2611 (d) 2501
  23. 23.If a + b = 17 and ab = 72, what is a² + b²? (a) 217 (b) 289 (c) 145 (d) 433
  24. 24.What is the sum of the roots of x² − 10x + 24 = 0? (a) 11 (b) 10 (c) 24 (d) 9
  25. 25.What is the product of the roots of x² − 12x + 32 = 0? (a) 12 (b) 33 (c) 32 (d) 31
  26. 26.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 8) (b) (x − 2)(x − 4) (c) (x + 1)(x + 5) (d) (x + 2)(x + 4)
  27. 27.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x + 1)(x + 72) (c) (x + 1)(x + 16) (d) (x − 8)(x − 9)
  28. 28.If a + b = 15 and ab = 56, what is (a − b)²? (a) 225 (b) 169 (c) 1 (d) 113
  29. 29.What is the sum of the roots of x² − 14x + 45 = 0? (a) 14 (b) 13 (c) 15 (d) 45
  30. 30.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 32) (b) (x + 4)(x + 8) (c) (x + 1)(x + 11) (d) (x − 4)(x − 8)
  31. 31.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 2025 (b) 3035 (c) 2525 (d) 3025
  32. 32.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 4) (b) (x − 2)(x − 3) (c) (x + 2)(x + 3) (d) (x + 1)(x + 6)
  33. 33.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 5051 (b) 4901 (c) 4761 (d) 5041
  34. 34.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 28) (b) (x − 4)(x − 7) (c) (x + 1)(x + 10) (d) (x + 4)(x + 7)
  35. 35.What is the product of the roots of x² − 6x + 8 = 0? (a) 8 (b) 9 (c) 6 (d) 7
  36. 36.If a + b = 18 and ab = 80, what is a² + b²? (a) 164 (b) 244 (c) 324 (d) 484
  37. 37.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 9604 (b) 10404 (c) 10004 (d) 10414
  38. 38.If a + b = 18 and ab = 80, what is (a − b)²? (a) 2 (b) 164 (c) 4 (d) 324
  39. 39.If a + b = 20 and ab = 99, what is (a − b)²? (a) 2 (b) 202 (c) 400 (d) 4
  40. 40.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2219 (b) 2209 (c) 2809 (d) 2491

Answer key

  1. (x + 6)(x + 7)
  2. 10201
  3. 1024
  4. (x + 2)(x + 6)
  5. 125
  6. 16
  7. 1
  8. 625
  9. 6724
  10. 117
  11. 6 and 7
  12. 9604
  13. 56
  14. (x + 3)(x + 9)
  15. 9
  16. 1225
  17. 8
  18. (x + 5)(x + 7)
  19. 10
  20. 4 and 9
  21. 82
  22. 2601
  23. 145
  24. 10
  25. 32
  26. (x + 2)(x + 4)
  27. (x + 8)(x + 9)
  28. 1
  29. 14
  30. (x + 4)(x + 8)
  31. 3025
  32. (x + 2)(x + 3)
  33. 5041
  34. (x + 4)(x + 7)
  35. 8
  36. 164
  37. 10404
  38. 4
  39. 4
  40. 2209

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