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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 495 · 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 + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 5)(x + 6) (c) (x + 1)(x + 30) (d) (x + 1)(x + 10)
  2. 2.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 896 (b) 1024 (c) 784 (d) 794
  3. 3.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)
  4. 4.Which is the correct factorisation of x² + 10x + 24? (a) (x − 4)(x − 6) (b) (x + 1)(x + 24) (c) (x + 1)(x + 9) (d) (x + 4)(x + 6)
  5. 5.What are the roots of x² − 9x + 20 = 0? (a) 4 and 5 (b) 1 and 20 (c) 1 and 8 (d) 2 and 10
  6. 6.If a + b = 5 and ab = 6, what is a² + b²? (a) 25 (b) 19 (c) 13 (d) 37
  7. 7.What is the product of the roots of x² − 7x + 10 = 0? (a) 10 (b) 11 (c) 7 (d) 9
  8. 8.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 235 (b) 625 (c) 375 (d) 225
  9. 9.If a + b = 10 and ab = 24, what is (a − b)²? (a) 52 (b) 4 (c) 100 (d) 2
  10. 10.If a + b = 4 and ab = 3, what is a² + b²? (a) 13 (b) 10 (c) 22 (d) 16
  11. 11.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 409 (b) 539 (c) 289 (d) 529
  12. 12.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 14 (c) 16 (d) 15
  13. 13.What is the sum of the roots of x² − 15x + 54 = 0? (a) 15 (b) 14 (c) 54 (d) 16
  14. 14.If a + b = 14 and ab = 48, what is a² + b²? (a) 292 (b) 196 (c) 100 (d) 148
  15. 15.If a + b = 19 and ab = 90, what is a² + b²? (a) 361 (b) 271 (c) 181 (d) 541
  16. 16.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 32) (b) (x − 4)(x − 8) (c) (x + 4)(x + 8) (d) (x + 1)(x + 11)
  17. 17.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6401 (b) 6571 (c) 6241 (d) 6561
  18. 18.If a + b = 13 and ab = 30, what is (a − b)²? (a) 109 (b) 169 (c) 7 (d) 49
  19. 19.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 1024 (b) 784 (c) 904 (d) 1034
  20. 20.If a + b = 14 and ab = 24, what is a² + b²? (a) 172 (b) 148 (c) 196 (d) 244
  21. 21.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 4) (b) (x + 1)(x + 6) (c) (x + 2)(x + 3) (d) (x − 2)(x − 3)
  22. 22.Which is the correct factorisation of x² + 12x + 27? (a) (x − 3)(x − 9) (b) (x + 1)(x + 27) (c) (x + 3)(x + 9) (d) (x + 1)(x + 11)
  23. 23.What are the roots of x² − 11x + 30 = 0? (a) 1 and 30 (b) 1 and 10 (c) 2 and 15 (d) 5 and 6
  24. 24.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4771 (b) 5041 (c) 4761 (d) 4899
  25. 25.Which is the correct factorisation of x² + 9x + 14? (a) (x − 2)(x − 7) (b) (x + 1)(x + 14) (c) (x + 1)(x + 8) (d) (x + 2)(x + 7)
  26. 26.If a + b = 21 and ab = 108, what is a² + b²? (a) 225 (b) 657 (c) 441 (d) 333
  27. 27.What is the sum of the roots of x² − 17x + 72 = 0? (a) 17 (b) 18 (c) 72 (d) 16
  28. 28.If a + b = 20 and ab = 96, what is a² + b²? (a) 304 (b) 400 (c) 208 (d) 592
  29. 29.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2401 (b) 2601 (c) 2499 (d) 2411
  30. 30.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 8096 (b) 7744 (c) 7754 (d) 8464
  31. 31.If a + b = 13 and ab = 40, what is (a − b)²? (a) 169 (b) 9 (c) 89 (d) 3
  32. 32.What is the sum of the roots of x² − 12x + 35 = 0? (a) 35 (b) 13 (c) 11 (d) 12
  33. 33.What is the product of the roots of x² − 13x + 40 = 0? (a) 39 (b) 40 (c) 41 (d) 13
  34. 34.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 11035 (b) 10025 (c) 9025 (d) 11025
  35. 35.What is the sum of the roots of x² − 16x + 63 = 0? (a) 17 (b) 15 (c) 63 (d) 16
  36. 36.If a + b = 16 and ab = 55, what is a² + b²? (a) 366 (b) 256 (c) 146 (d) 201
  37. 37.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 875 (b) 625 (c) 635 (d) 1225
  38. 38.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1296 (b) 1946 (c) 1936 (d) 1616
  39. 39.If a + b = 12 and ab = 11, what is (a − b)²? (a) 10 (b) 144 (c) 100 (d) 122
  40. 40.If a + b = 20 and ab = 96, what is (a − b)²? (a) 208 (b) 400 (c) 4 (d) 16

Answer key

  1. (x + 5)(x + 6)
  2. 784
  3. (x + 2)(x + 6)
  4. (x + 4)(x + 6)
  5. 4 and 5
  6. 13
  7. 10
  8. 225
  9. 4
  10. 10
  11. 529
  12. 15
  13. 15
  14. 100
  15. 181
  16. (x + 4)(x + 8)
  17. 6561
  18. 49
  19. 1024
  20. 148
  21. (x + 2)(x + 3)
  22. (x + 3)(x + 9)
  23. 5 and 6
  24. 4761
  25. (x + 2)(x + 7)
  26. 225
  27. 17
  28. 208
  29. 2401
  30. 7744
  31. 9
  32. 12
  33. 40
  34. 11025
  35. 16
  36. 146
  37. 625
  38. 1936
  39. 100
  40. 16

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