← Identities, polynomials and quadratic roots lesson New set →

Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 561 · 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 + 13x + 40? (a) (x − 5)(x − 8) (b) (x + 5)(x + 8) (c) (x + 1)(x + 12) (d) (x + 1)(x + 40)
  2. 2.If a + b = 15 and ab = 54, what is a² + b²? (a) 225 (b) 171 (c) 117 (d) 333
  3. 3.If a + b = 13 and ab = 22, what is (a − b)²? (a) 169 (b) 81 (c) 125 (d) 9
  4. 4.If a + b = 11 and ab = 10, what is a² + b²? (a) 111 (b) 141 (c) 101 (d) 121
  5. 5.If a + b = 20 and ab = 99, what is (a − b)²? (a) 202 (b) 4 (c) 400 (d) 2
  6. 6.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 10) (b) (x + 4)(x + 7) (c) (x + 1)(x + 28) (d) (x − 4)(x − 7)
  7. 7.Which is the correct factorisation of x² + 7x + 10? (a) (x + 2)(x + 5) (b) (x − 2)(x − 5) (c) (x + 1)(x + 10) (d) (x + 1)(x + 6)
  8. 8.Which is the correct factorisation of x² + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 1)(x + 10) (c) (x + 1)(x + 30) (d) (x + 5)(x + 6)
  9. 9.If a + b = 6 and ab = 8, what is a² + b²? (a) 52 (b) 28 (c) 36 (d) 20
  10. 10.What is the sum of the roots of x² − 5x + 6 = 0? (a) 4 (b) 7 (c) 6 (d) 5
  11. 11.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 10201 (b) 9801 (c) 9999 (d) 9811
  12. 12.What are the roots of x² − 11x + 30 = 0? (a) 5 and 6 (b) 2 and 15 (c) 1 and 10 (d) 1 and 30
  13. 13.If a + b = 13 and ab = 30, what is a² + b²? (a) 169 (b) 109 (c) 139 (d) 229
  14. 14.Which is the correct factorisation of x² + 13x + 42? (a) (x + 6)(x + 7) (b) (x + 1)(x + 12) (c) (x + 1)(x + 42) (d) (x − 6)(x − 7)
  15. 15.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 1)(x + 13) (c) (x + 1)(x + 48) (d) (x + 6)(x + 8)
  16. 16.What is the product of the roots of x² − 5x + 6 = 0? (a) 5 (b) 6 (c) 7 (d) 8
  17. 17.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 6) (b) (x + 2)(x + 3) (c) (x − 2)(x − 3) (d) (x + 1)(x + 4)
  18. 18.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6399 (b) 6241 (c) 6561 (d) 6251
  19. 19.If a + b = 12 and ab = 35, what is a² + b²? (a) 214 (b) 144 (c) 109 (d) 74
  20. 20.Which is the correct factorisation of x² + 9x + 18? (a) (x − 3)(x − 6) (b) (x + 1)(x + 18) (c) (x + 1)(x + 8) (d) (x + 3)(x + 6)
  21. 21.If a + b = 12 and ab = 35, what is (a − b)²? (a) 2 (b) 4 (c) 74 (d) 144
  22. 22.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9604 (b) 10404 (c) 9614 (d) 9996
  23. 23.If a + b = 7 and ab = 12, what is (a − b)²? (a) 25 (b) 49 (c) 37 (d) 1
  24. 24.If a + b = 8 and ab = 7, what is a² + b²? (a) 50 (b) 57 (c) 64 (d) 78
  25. 25.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 1 and 16 (c) 2 and 36 (d) 8 and 9
  26. 26.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2611 (b) 2501 (c) 2601 (d) 2401
  27. 27.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 225 (b) 375 (c) 625 (d) 235
  28. 28.What are the roots of x² − 10x + 24 = 0? (a) 1 and 9 (b) 1 and 24 (c) 2 and 12 (d) 4 and 6
  29. 29.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x − 8)(x − 9) (c) (x + 8)(x + 9) (d) (x + 1)(x + 72)
  30. 30.What is the product of the roots of x² − 12x + 27 = 0? (a) 26 (b) 28 (c) 27 (d) 12
  31. 31.If a + b = 16 and ab = 60, what is a² + b²? (a) 376 (b) 136 (c) 196 (d) 256
  32. 32.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 901 (b) 841 (c) 971 (d) 961
  33. 33.What is the sum of the roots of x² − 15x + 54 = 0? (a) 16 (b) 14 (c) 15 (d) 54
  34. 34.Which is the correct factorisation of x² + 13x + 36? (a) (x + 4)(x + 9) (b) (x + 1)(x + 36) (c) (x + 1)(x + 12) (d) (x − 4)(x − 9)
  35. 35.What are the roots of x² − 8x + 15 = 0? (a) 2 and 6 (b) 1 and 15 (c) 1 and 7 (d) 3 and 5
  36. 36.What is the product of the roots of x² − 9x + 14 = 0? (a) 14 (b) 15 (c) 13 (d) 9
  37. 37.If a + b = 15 and ab = 44, what is (a − b)²? (a) 7 (b) 225 (c) 49 (d) 137
  38. 38.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 15 (c) 16 (d) 14
  39. 39.If a + b = 12 and ab = 27, what is a² + b²? (a) 198 (b) 90 (c) 144 (d) 117
  40. 40.If a + b = 8 and ab = 7, what is (a − b)²? (a) 36 (b) 50 (c) 64 (d) 6

Answer key

  1. (x + 5)(x + 8)
  2. 117
  3. 81
  4. 101
  5. 4
  6. (x + 4)(x + 7)
  7. (x + 2)(x + 5)
  8. (x + 5)(x + 6)
  9. 20
  10. 5
  11. 9801
  12. 5 and 6
  13. 109
  14. (x + 6)(x + 7)
  15. (x + 6)(x + 8)
  16. 6
  17. (x + 2)(x + 3)
  18. 6241
  19. 74
  20. (x + 3)(x + 6)
  21. 4
  22. 9604
  23. 1
  24. 50
  25. 8 and 9
  26. 2601
  27. 225
  28. 4 and 6
  29. (x + 8)(x + 9)
  30. 27
  31. 136
  32. 961
  33. 15
  34. (x + 4)(x + 9)
  35. 3 and 5
  36. 14
  37. 49
  38. 15
  39. 90
  40. 36

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