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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 223 · 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 = 20 and ab = 99, what is (a − b)²? (a) 4 (b) 202 (c) 400 (d) 2
  2. 2.What is the sum of the roots of x² − 8x + 12 = 0? (a) 9 (b) 8 (c) 7 (d) 12
  3. 3.If a + b = 20 and ab = 96, what is (a − b)²? (a) 4 (b) 208 (c) 16 (d) 400
  4. 4.Which is the correct factorisation of x² + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 1)(x + 6) (c) (x + 1)(x + 12) (d) (x + 3)(x + 4)
  5. 5.If a + b = 15 and ab = 54, what is a² + b²? (a) 171 (b) 333 (c) 117 (d) 225
  6. 6.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 40) (b) (x − 5)(x − 8) (c) (x + 5)(x + 8) (d) (x + 1)(x + 12)
  7. 7.If a + b = 9 and ab = 18, what is a² + b²? (a) 45 (b) 81 (c) 63 (d) 117
  8. 8.What are the roots of x² − 15x + 56 = 0? (a) 7 and 8 (b) 1 and 14 (c) 1 and 56 (d) 2 and 28
  9. 9.What is the product of the roots of x² − 7x + 10 = 0? (a) 9 (b) 7 (c) 11 (d) 10
  10. 10.What is the product of the roots of x² − 11x + 24 = 0? (a) 24 (b) 25 (c) 23 (d) 11
  11. 11.If a + b = 16 and ab = 63, what is a² + b²? (a) 193 (b) 256 (c) 382 (d) 130
  12. 12.Which is the correct factorisation of x² + 12x + 27? (a) (x + 3)(x + 9) (b) (x − 3)(x − 9) (c) (x + 1)(x + 27) (d) (x + 1)(x + 11)
  13. 13.If a + b = 14 and ab = 24, what is a² + b²? (a) 196 (b) 172 (c) 244 (d) 148
  14. 14.If a + b = 5 and ab = 4, what is a² + b²? (a) 17 (b) 33 (c) 25 (d) 21
  15. 15.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 5) (b) (x − 2)(x − 4) (c) (x + 1)(x + 8) (d) (x + 2)(x + 4)
  16. 16.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 9409 (b) 9419 (c) 10609 (d) 9991
  17. 17.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 7066 (b) 5776 (c) 6416 (d) 7056
  18. 18.What is the product of the roots of x² − 9x + 18 = 0? (a) 17 (b) 18 (c) 9 (d) 19
  19. 19.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 8 and 9 (c) 1 and 16 (d) 2 and 36
  20. 20.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 3025 (b) 2025 (c) 3035 (d) 2525
  21. 21.What is the sum of the roots of x² − 8x + 15 = 0? (a) 8 (b) 15 (c) 7 (d) 9
  22. 22.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2509 (b) 2819 (c) 2809 (d) 2209
  23. 23.If a + b = 21 and ab = 108, what is (a − b)²? (a) 225 (b) 3 (c) 9 (d) 441
  24. 24.If a + b = 17 and ab = 66, what is a² + b²? (a) 421 (b) 289 (c) 157 (d) 223
  25. 25.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x + 1)(x + 15) (c) (x + 1)(x + 63) (d) (x − 7)(x − 9)
  26. 26.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 4225 (b) 4925 (c) 5625 (d) 5635
  27. 27.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x + 7)(x + 8) (c) (x − 7)(x − 8) (d) (x + 1)(x + 56)
  28. 28.If a + b = 18 and ab = 72, what is a² + b²? (a) 180 (b) 324 (c) 252 (d) 468
  29. 29.If a + b = 19 and ab = 90, what is a² + b²? (a) 181 (b) 541 (c) 271 (d) 361
  30. 30.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)
  31. 31.If a + b = 7 and ab = 10, what is a² + b²? (a) 49 (b) 29 (c) 39 (d) 69
  32. 32.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x + 6)(x + 9) (c) (x − 6)(x − 9) (d) (x + 1)(x + 14)
  33. 33.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 409 (b) 289 (c) 539 (d) 529
  34. 34.If a + b = 15 and ab = 56, what is a² + b²? (a) 169 (b) 225 (c) 113 (d) 337
  35. 35.If a + b = 5 and ab = 4, what is (a − b)²? (a) 17 (b) 3 (c) 25 (d) 9
  36. 36.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 625 (b) 235 (c) 225 (d) 375
  37. 37.What are the roots of x² − 15x + 54 = 0? (a) 1 and 54 (b) 1 and 14 (c) 2 and 27 (d) 6 and 9
  38. 38.If a + b = 13 and ab = 30, what is (a − b)²? (a) 7 (b) 109 (c) 49 (d) 169
  39. 39.If a + b = 18 and ab = 80, what is a² + b²? (a) 164 (b) 484 (c) 244 (d) 324
  40. 40.If a + b = 20 and ab = 96, what is a² + b²? (a) 592 (b) 304 (c) 208 (d) 400

Answer key

  1. 4
  2. 8
  3. 16
  4. (x + 3)(x + 4)
  5. 117
  6. (x + 5)(x + 8)
  7. 45
  8. 7 and 8
  9. 10
  10. 24
  11. 130
  12. (x + 3)(x + 9)
  13. 148
  14. 17
  15. (x + 2)(x + 4)
  16. 9409
  17. 7056
  18. 18
  19. 8 and 9
  20. 3025
  21. 8
  22. 2809
  23. 9
  24. 157
  25. (x + 7)(x + 9)
  26. 5625
  27. (x + 7)(x + 8)
  28. 180
  29. 181
  30. (x + 5)(x + 9)
  31. 29
  32. (x + 6)(x + 9)
  33. 529
  34. 113
  35. 9
  36. 225
  37. 6 and 9
  38. 49
  39. 164
  40. 208

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