← Identities, polynomials and quadratic roots lesson New set →

Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 870 · 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 48²? (a) 2304 (b) 2314 (c) 2704 (d) 2496
  2. 2.What is the sum of the roots of x² − 13x + 40 = 0? (a) 14 (b) 13 (c) 12 (d) 40
  3. 3.If a + b = 13 and ab = 42, what is (a − b)²? (a) 85 (b) 169 (c) 1 (d) 127
  4. 4.If a + b = 8 and ab = 15, what is a² + b²? (a) 49 (b) 94 (c) 34 (d) 64
  5. 5.What is the sum of the roots of x² − 14x + 48 = 0? (a) 48 (b) 15 (c) 14 (d) 13
  6. 6.Which is the correct factorisation of x² + 12x + 32? (a) (x − 4)(x − 8) (b) (x + 4)(x + 8) (c) (x + 1)(x + 32) (d) (x + 1)(x + 11)
  7. 7.What is the sum of the roots of x² − 17x + 72 = 0? (a) 17 (b) 72 (c) 16 (d) 18
  8. 8.What is the sum of the roots of x² − 11x + 18 = 0? (a) 11 (b) 12 (c) 18 (d) 10
  9. 9.What are the roots of x² − 13x + 40 = 0? (a) 1 and 12 (b) 5 and 8 (c) 2 and 20 (d) 1 and 40
  10. 10.If a + b = 7 and ab = 10, what is a² + b²? (a) 49 (b) 69 (c) 39 (d) 29
  11. 11.If a + b = 20 and ab = 96, what is a² + b²? (a) 304 (b) 208 (c) 400 (d) 592
  12. 12.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 5184 (b) 4624 (c) 4634 (d) 4896
  13. 13.If a + b = 11 and ab = 24, what is a² + b²? (a) 97 (b) 121 (c) 73 (d) 169
  14. 14.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 7569 (b) 8649 (c) 8109 (d) 8659
  15. 15.If a + b = 3 and ab = 2, what is (a − b)²? (a) 9 (b) 1 (c) 5 (d) 7
  16. 16.If a + b = 16 and ab = 63, what is a² + b²? (a) 193 (b) 382 (c) 130 (d) 256
  17. 17.Which is the correct factorisation of x² + 13x + 36? (a) (x + 4)(x + 9) (b) (x + 1)(x + 12) (c) (x − 4)(x − 9) (d) (x + 1)(x + 36)
  18. 18.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1089 (b) 909 (c) 729 (d) 1099
  19. 19.If a + b = 12 and ab = 27, what is a² + b²? (a) 117 (b) 90 (c) 198 (d) 144
  20. 20.If a + b = 15 and ab = 54, what is a² + b²? (a) 333 (b) 225 (c) 117 (d) 171
  21. 21.What is the sum of the roots of x² − 10x + 21 = 0? (a) 9 (b) 21 (c) 10 (d) 11
  22. 22.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x + 1)(x + 56) (c) (x − 7)(x − 8) (d) (x + 7)(x + 8)
  23. 23.If a + b = 19 and ab = 90, what is a² + b²? (a) 361 (b) 181 (c) 541 (d) 271
  24. 24.What are the roots of x² − 10x + 21 = 0? (a) 2 and 8 (b) 3 and 7 (c) 1 and 21 (d) 1 and 9
  25. 25.If a + b = 16 and ab = 48, what is a² + b²? (a) 256 (b) 208 (c) 352 (d) 160
  26. 26.What is the product of the roots of x² − 15x + 56 = 0? (a) 55 (b) 15 (c) 57 (d) 56
  27. 27.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6241 (b) 6561 (c) 6401 (d) 6571
  28. 28.What is the sum of the roots of x² − 6x + 8 = 0? (a) 8 (b) 6 (c) 5 (d) 7
  29. 29.If a + b = 19 and ab = 88, what is a² + b²? (a) 537 (b) 273 (c) 361 (d) 185
  30. 30.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)
  31. 31.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6251 (b) 6399 (c) 6241 (d) 6561
  32. 32.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 841 (b) 901 (c) 961 (d) 971
  33. 33.Which is the correct factorisation of x² + 13x + 40? (a) (x − 5)(x − 8) (b) (x + 5)(x + 8) (c) (x + 1)(x + 12) (d) (x + 1)(x + 40)
  34. 34.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 8096 (b) 7744 (c) 8464 (d) 7754
  35. 35.If a + b = 11 and ab = 24, what is (a − b)²? (a) 121 (b) 25 (c) 5 (d) 73
  36. 36.If a + b = 7 and ab = 6, what is a² + b²? (a) 43 (b) 37 (c) 61 (d) 49
  37. 37.If a + b = 7 and ab = 10, what is (a − b)²? (a) 29 (b) 3 (c) 9 (d) 49
  38. 38.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 9) (b) (x − 2)(x − 8) (c) (x + 1)(x + 16) (d) (x + 2)(x + 8)
  39. 39.What is the product of the roots of x² − 9x + 18 = 0? (a) 19 (b) 9 (c) 18 (d) 17
  40. 40.If a + b = 17 and ab = 60, what is (a − b)²? (a) 169 (b) 49 (c) 289 (d) 7

Answer key

  1. 2304
  2. 13
  3. 1
  4. 34
  5. 14
  6. (x + 4)(x + 8)
  7. 17
  8. 11
  9. 5 and 8
  10. 29
  11. 208
  12. 4624
  13. 73
  14. 8649
  15. 1
  16. 130
  17. (x + 4)(x + 9)
  18. 1089
  19. 90
  20. 117
  21. 10
  22. (x + 7)(x + 8)
  23. 181
  24. 3 and 7
  25. 160
  26. 56
  27. 6561
  28. 6
  29. 185
  30. (x + 4)(x + 6)
  31. 6241
  32. 961
  33. (x + 5)(x + 8)
  34. 7744
  35. 25
  36. 37
  37. 9
  38. (x + 2)(x + 8)
  39. 18
  40. 49

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