← Identities, polynomials and quadratic roots lesson New set →

Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 239 · 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 + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 8)(x + 9) (c) (x + 1)(x + 72) (d) (x − 8)(x − 9)
  2. 2.What is the product of the roots of x² − 14x + 45 = 0? (a) 46 (b) 44 (c) 45 (d) 14
  3. 3.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 576 (b) 416 (c) 256 (d) 586
  4. 4.If a + b = 14 and ab = 24, what is a² + b²? (a) 196 (b) 244 (c) 148 (d) 172
  5. 5.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 24) (b) (x + 1)(x + 10) (c) (x + 3)(x + 8) (d) (x − 3)(x − 8)
  6. 6.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 5) (b) (x + 1)(x + 8) (c) (x + 2)(x + 4) (d) (x − 2)(x − 4)
  7. 7.What are the roots of x² − 17x + 72 = 0? (a) 2 and 36 (b) 1 and 72 (c) 1 and 16 (d) 8 and 9
  8. 8.If a + b = 7 and ab = 12, what is (a − b)²? (a) 37 (b) 49 (c) 1 (d) 25
  9. 9.If a + b = 14 and ab = 33, what is a² + b²? (a) 196 (b) 262 (c) 130 (d) 163
  10. 10.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 12) (b) (x − 5)(x − 8) (c) (x + 1)(x + 40) (d) (x + 5)(x + 8)
  11. 11.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 12) (b) (x − 4)(x − 9) (c) (x + 4)(x + 9) (d) (x + 1)(x + 36)
  12. 12.Which is the correct factorisation of x² + 13x + 42? (a) (x + 6)(x + 7) (b) (x + 1)(x + 42) (c) (x + 1)(x + 12) (d) (x − 6)(x − 7)
  13. 13.What are the roots of x² − 15x + 54 = 0? (a) 6 and 9 (b) 2 and 27 (c) 1 and 54 (d) 1 and 14
  14. 14.What is the sum of the roots of x² − 15x + 54 = 0? (a) 15 (b) 14 (c) 16 (d) 54
  15. 15.If a + b = 10 and ab = 24, what is (a − b)²? (a) 52 (b) 100 (c) 2 (d) 4
  16. 16.If a + b = 9 and ab = 20, what is a² + b²? (a) 61 (b) 121 (c) 41 (d) 81
  17. 17.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 7235 (b) 9025 (c) 7225 (d) 8075
  18. 18.What is the product of the roots of x² − 7x + 12 = 0? (a) 7 (b) 13 (c) 11 (d) 12
  19. 19.If a + b = 18 and ab = 80, what is a² + b²? (a) 484 (b) 244 (c) 324 (d) 164
  20. 20.If a + b = 19 and ab = 90, what is a² + b²? (a) 541 (b) 361 (c) 271 (d) 181
  21. 21.Which is the correct factorisation of x² + 10x + 16? (a) (x + 2)(x + 8) (b) (x + 1)(x + 16) (c) (x − 2)(x − 8) (d) (x + 1)(x + 9)
  22. 22.What is the sum of the roots of x² − 13x + 36 = 0? (a) 12 (b) 13 (c) 36 (d) 14
  23. 23.What is the product of the roots of x² − 7x + 10 = 0? (a) 11 (b) 9 (c) 10 (d) 7
  24. 24.What is the sum of the roots of x² − 9x + 20 = 0? (a) 10 (b) 20 (c) 8 (d) 9
  25. 25.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 4225 (b) 3035 (c) 3025 (d) 3575
  26. 26.If a + b = 5 and ab = 4, what is (a − b)²? (a) 17 (b) 25 (c) 9 (d) 3
  27. 27.What is the sum of the roots of x² − 12x + 32 = 0? (a) 11 (b) 13 (c) 12 (d) 32
  28. 28.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 1)(x + 13) (c) (x + 6)(x + 8) (d) (x + 1)(x + 48)
  29. 29.If a + b = 11 and ab = 10, what is (a − b)²? (a) 101 (b) 121 (c) 81 (d) 9
  30. 30.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4899 (b) 5041 (c) 4771 (d) 4761
  31. 31.What are the roots of x² − 5x + 6 = 0? (a) 2 and 4 (b) 1 and 6 (c) 1 and 4 (d) 2 and 3
  32. 32.If a + b = 14 and ab = 40, what is (a − b)²? (a) 116 (b) 196 (c) 36 (d) 6
  33. 33.If a + b = 15 and ab = 36, what is (a − b)²? (a) 153 (b) 225 (c) 9 (d) 81
  34. 34.If a + b = 6 and ab = 8, what is (a − b)²? (a) 20 (b) 4 (c) 36 (d) 2
  35. 35.If a + b = 14 and ab = 40, what is a² + b²? (a) 116 (b) 156 (c) 276 (d) 196
  36. 36.If a + b = 12 and ab = 35, what is a² + b²? (a) 109 (b) 144 (c) 214 (d) 74
  37. 37.What is the product of the roots of x² − 13x + 40 = 0? (a) 13 (b) 40 (c) 39 (d) 41
  38. 38.If a + b = 12 and ab = 20, what is a² + b²? (a) 144 (b) 104 (c) 184 (d) 124
  39. 39.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 884 (b) 1156 (c) 676 (d) 686
  40. 40.If a + b = 17 and ab = 66, what is a² + b²? (a) 223 (b) 157 (c) 289 (d) 421

Answer key

  1. (x + 8)(x + 9)
  2. 45
  3. 576
  4. 148
  5. (x + 3)(x + 8)
  6. (x + 2)(x + 4)
  7. 8 and 9
  8. 1
  9. 130
  10. (x + 5)(x + 8)
  11. (x + 4)(x + 9)
  12. (x + 6)(x + 7)
  13. 6 and 9
  14. 15
  15. 4
  16. 41
  17. 7225
  18. 12
  19. 164
  20. 181
  21. (x + 2)(x + 8)
  22. 13
  23. 10
  24. 9
  25. 3025
  26. 9
  27. 12
  28. (x + 6)(x + 8)
  29. 81
  30. 4761
  31. 2 and 3
  32. 36
  33. 81
  34. 4
  35. 116
  36. 74
  37. 40
  38. 104
  39. 676
  40. 157

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