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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 354 · 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 + 10x + 24? (a) (x + 1)(x + 9) (b) (x + 1)(x + 24) (c) (x − 4)(x − 6) (d) (x + 4)(x + 6)
  2. 2.If a + b = 15 and ab = 36, what is a² + b²? (a) 225 (b) 297 (c) 153 (d) 189
  3. 3.If a + b = 6 and ab = 5, what is a² + b²? (a) 31 (b) 36 (c) 46 (d) 26
  4. 4.If a + b = 19 and ab = 84, what is (a − b)²? (a) 193 (b) 361 (c) 25 (d) 5
  5. 5.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10004 (b) 9604 (c) 10404 (d) 10414
  6. 6.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 13) (b) (x + 5)(x + 9) (c) (x + 1)(x + 45) (d) (x − 5)(x − 9)
  7. 7.What is the sum of the roots of x² − 11x + 24 = 0? (a) 12 (b) 10 (c) 11 (d) 24
  8. 8.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 21) (b) (x + 1)(x + 9) (c) (x + 3)(x + 7) (d) (x − 3)(x − 7)
  9. 9.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 10609 (b) 9409 (c) 9991 (d) 9419
  10. 10.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 256 (b) 576 (c) 384 (d) 266
  11. 11.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 35) (b) (x − 5)(x − 7) (c) (x + 1)(x + 11) (d) (x + 5)(x + 7)
  12. 12.What are the roots of x² − 15x + 54 = 0? (a) 6 and 9 (b) 2 and 27 (c) 1 and 14 (d) 1 and 54
  13. 13.If a + b = 9 and ab = 8, what is a² + b²? (a) 97 (b) 73 (c) 81 (d) 65
  14. 14.If a + b = 16 and ab = 60, what is a² + b²? (a) 136 (b) 196 (c) 376 (d) 256
  15. 15.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 6) (b) (x − 2)(x − 3) (c) (x + 2)(x + 3) (d) (x + 1)(x + 4)
  16. 16.If a + b = 20 and ab = 99, what is (a − b)²? (a) 202 (b) 2 (c) 400 (d) 4
  17. 17.If a + b = 15 and ab = 44, what is a² + b²? (a) 137 (b) 313 (c) 181 (d) 225
  18. 18.What is the product of the roots of x² − 11x + 30 = 0? (a) 11 (b) 29 (c) 31 (d) 30
  19. 19.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2484 (b) 2916 (c) 2126 (d) 2116
  20. 20.If a + b = 10 and ab = 21, what is (a − b)²? (a) 16 (b) 100 (c) 58 (d) 4
  21. 21.If a + b = 13 and ab = 42, what is (a − b)²? (a) 169 (b) 127 (c) 1 (d) 85
  22. 22.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 17 (c) 72 (d) 16
  23. 23.What is the sum of the roots of x² − 15x + 54 = 0? (a) 16 (b) 14 (c) 15 (d) 54
  24. 24.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 8075 (b) 9025 (c) 7235 (d) 7225
  25. 25.If a + b = 12 and ab = 20, what is a² + b²? (a) 124 (b) 184 (c) 104 (d) 144
  26. 26.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 10404 (b) 9604 (c) 9996 (d) 9614
  27. 27.What is the sum of the roots of x² − 12x + 32 = 0? (a) 13 (b) 32 (c) 12 (d) 11
  28. 28.If a + b = 16 and ab = 55, what is (a − b)²? (a) 36 (b) 146 (c) 6 (d) 256
  29. 29.What are the roots of x² − 11x + 28 = 0? (a) 2 and 14 (b) 4 and 7 (c) 1 and 10 (d) 1 and 28
  30. 30.What is the sum of the roots of x² − 15x + 56 = 0? (a) 16 (b) 14 (c) 56 (d) 15
  31. 31.If a + b = 13 and ab = 22, what is (a − b)²? (a) 169 (b) 125 (c) 9 (d) 81
  32. 32.What are the roots of x² − 13x + 42 = 0? (a) 1 and 12 (b) 2 and 21 (c) 6 and 7 (d) 1 and 42
  33. 33.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)
  34. 34.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x + 1)(x + 16) (c) (x − 8)(x − 9) (d) (x + 1)(x + 72)
  35. 35.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)
  36. 36.If a + b = 20 and ab = 96, what is a² + b²? (a) 400 (b) 592 (c) 304 (d) 208
  37. 37.Which is the correct factorisation of x² + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 1)(x + 6) (c) (x + 3)(x + 4) (d) (x + 1)(x + 12)
  38. 38.If a + b = 9 and ab = 20, what is (a − b)²? (a) 61 (b) 1 (c) 41 (d) 81
  39. 39.Which is the correct factorisation of x² + 8x + 12? (a) (x + 2)(x + 6) (b) (x + 1)(x + 7) (c) (x + 1)(x + 12) (d) (x − 2)(x − 6)
  40. 40.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 8084 (b) 7406 (c) 7396 (d) 8836

Answer key

  1. (x + 4)(x + 6)
  2. 153
  3. 26
  4. 25
  5. 10404
  6. (x + 5)(x + 9)
  7. 11
  8. (x + 3)(x + 7)
  9. 9409
  10. 256
  11. (x + 5)(x + 7)
  12. 6 and 9
  13. 65
  14. 136
  15. (x + 2)(x + 3)
  16. 4
  17. 137
  18. 30
  19. 2116
  20. 16
  21. 1
  22. 17
  23. 15
  24. 7225
  25. 104
  26. 9604
  27. 12
  28. 36
  29. 4 and 7
  30. 15
  31. 81
  32. 6 and 7
  33. (x + 7)(x + 9)
  34. (x + 8)(x + 9)
  35. (x + 6)(x + 7)
  36. 208
  37. (x + 3)(x + 4)
  38. 1
  39. (x + 2)(x + 6)
  40. 7396

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