← Identities, polynomials and quadratic roots lesson New set →

Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 615 · 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 87²? (a) 7579 (b) 8091 (c) 7569 (d) 8649
  2. 2.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 8836 (b) 7406 (c) 8084 (d) 7396
  3. 3.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 7)(x + 9) (c) (x − 7)(x − 9) (d) (x + 1)(x + 63)
  4. 4.What is the product of the roots of x² − 16x + 63 = 0? (a) 16 (b) 64 (c) 62 (d) 63
  5. 5.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 635 (b) 1225 (c) 625 (d) 875
  6. 6.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10609 (b) 10619 (c) 10009 (d) 9409
  7. 7.If a + b = 10 and ab = 16, what is a² + b²? (a) 84 (b) 68 (c) 100 (d) 132
  8. 8.If a + b = 15 and ab = 36, what is (a − b)²? (a) 81 (b) 9 (c) 153 (d) 225
  9. 9.What are the roots of x² − 13x + 36 = 0? (a) 2 and 18 (b) 1 and 12 (c) 1 and 36 (d) 4 and 9
  10. 10.What is the product of the roots of x² − 13x + 36 = 0? (a) 13 (b) 37 (c) 36 (d) 35
  11. 11.If a + b = 14 and ab = 40, what is (a − b)²? (a) 36 (b) 116 (c) 6 (d) 196
  12. 12.What are the roots of x² − 11x + 28 = 0? (a) 4 and 7 (b) 1 and 28 (c) 1 and 10 (d) 2 and 14
  13. 13.What is the sum of the roots of x² − 13x + 42 = 0? (a) 12 (b) 13 (c) 14 (d) 42
  14. 14.What is the product of the roots of x² − 5x + 6 = 0? (a) 7 (b) 6 (c) 8 (d) 5
  15. 15.What is the sum of the roots of x² − 13x + 40 = 0? (a) 14 (b) 12 (c) 40 (d) 13
  16. 16.What is the sum of the roots of x² − 8x + 15 = 0? (a) 8 (b) 7 (c) 9 (d) 15
  17. 17.What are the roots of x² − 6x + 8 = 0? (a) 1 and 8 (b) 2 and 4 (c) 1 and 5 (d) 2 and 5
  18. 18.What are the roots of x² − 9x + 14 = 0? (a) 1 and 8 (b) 1 and 14 (c) 3 and 6 (d) 2 and 7
  19. 19.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8474 (b) 8104 (c) 8464 (d) 7744
  20. 20.If a + b = 20 and ab = 96, what is (a − b)²? (a) 208 (b) 4 (c) 16 (d) 400
  21. 21.If a + b = 21 and ab = 108, what is a² + b²? (a) 657 (b) 441 (c) 333 (d) 225
  22. 22.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)
  23. 23.What is the sum of the roots of x² − 14x + 48 = 0? (a) 13 (b) 15 (c) 14 (d) 48
  24. 24.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x + 6)(x + 8) (c) (x + 1)(x + 13) (d) (x − 6)(x − 8)
  25. 25.What is the sum of the roots of x² − 7x + 12 = 0? (a) 7 (b) 12 (c) 8 (d) 6
  26. 26.If a + b = 17 and ab = 72, what is (a − b)²? (a) 145 (b) 289 (c) 1 (d) 217
  27. 27.If a + b = 12 and ab = 32, what is (a − b)²? (a) 4 (b) 80 (c) 144 (d) 16
  28. 28.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3249 (b) 3969 (c) 3259 (d) 3591
  29. 29.If a + b = 15 and ab = 54, what is a² + b²? (a) 171 (b) 117 (c) 225 (d) 333
  30. 30.What is the sum of the roots of x² − 15x + 54 = 0? (a) 16 (b) 54 (c) 14 (d) 15
  31. 31.Which is the correct factorisation of x² + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 3)(x + 4) (c) (x + 1)(x + 12) (d) (x + 1)(x + 6)
  32. 32.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3731 (b) 3721 (c) 3481 (d) 3601
  33. 33.Which is the correct factorisation of x² + 8x + 15? (a) (x + 3)(x + 5) (b) (x + 1)(x + 7) (c) (x + 1)(x + 15) (d) (x − 3)(x − 5)
  34. 34.What are the roots of x² − 10x + 21 = 0? (a) 1 and 21 (b) 1 and 9 (c) 3 and 7 (d) 2 and 8
  35. 35.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 16) (b) (x + 2)(x + 8) (c) (x + 1)(x + 9) (d) (x − 2)(x − 8)
  36. 36.If a + b = 10 and ab = 21, what is (a − b)²? (a) 58 (b) 100 (c) 4 (d) 16
  37. 37.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3374 (b) 3596 (c) 3844 (d) 3364
  38. 38.What is the sum of the roots of x² − 14x + 45 = 0? (a) 13 (b) 14 (c) 15 (d) 45
  39. 39.What is the product of the roots of x² − 15x + 56 = 0? (a) 56 (b) 55 (c) 15 (d) 57
  40. 40.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 6)(x + 7) (c) (x + 1)(x + 12) (d) (x + 1)(x + 42)

Answer key

  1. 7569
  2. 7396
  3. (x + 7)(x + 9)
  4. 63
  5. 625
  6. 10609
  7. 68
  8. 81
  9. 4 and 9
  10. 36
  11. 36
  12. 4 and 7
  13. 13
  14. 6
  15. 13
  16. 8
  17. 2 and 4
  18. 2 and 7
  19. 8464
  20. 16
  21. 225
  22. (x + 4)(x + 6)
  23. 14
  24. (x + 6)(x + 8)
  25. 7
  26. 1
  27. 16
  28. 3249
  29. 117
  30. 15
  31. (x + 3)(x + 4)
  32. 3721
  33. (x + 3)(x + 5)
  34. 3 and 7
  35. (x + 2)(x + 8)
  36. 16
  37. 3364
  38. 14
  39. 56
  40. (x + 6)(x + 7)

Free maths olympiad lessons, practice and worksheets at talentjr.in/maths-olympiad