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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 163 · 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.What is the product of the roots of x² − 11x + 30 = 0? (a) 29 (b) 11 (c) 30 (d) 31
  2. 2.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 9) (b) (x + 1)(x + 21) (c) (x + 3)(x + 7) (d) (x − 3)(x − 7)
  3. 3.If a + b = 21 and ab = 108, what is a² + b²? (a) 333 (b) 441 (c) 225 (d) 657
  4. 4.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 9025 (b) 10025 (c) 11025 (d) 11035
  5. 5.What is the sum of the roots of x² − 16x + 63 = 0? (a) 17 (b) 16 (c) 63 (d) 15
  6. 6.What is the product of the roots of x² − 13x + 36 = 0? (a) 37 (b) 13 (c) 35 (d) 36
  7. 7.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x − 6)(x − 7) (c) (x + 1)(x + 12) (d) (x + 6)(x + 7)
  8. 8.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 361 (b) 401 (c) 451 (d) 441
  9. 9.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 11) (b) (x + 1)(x + 32) (c) (x − 4)(x − 8) (d) (x + 4)(x + 8)
  10. 10.What is the product of the roots of x² − 14x + 45 = 0? (a) 14 (b) 45 (c) 46 (d) 44
  11. 11.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2809 (b) 2209 (c) 2819 (d) 2509
  12. 12.If a + b = 9 and ab = 8, what is (a − b)²? (a) 7 (b) 65 (c) 81 (d) 49
  13. 13.If a + b = 14 and ab = 40, what is (a − b)²? (a) 196 (b) 6 (c) 36 (d) 116
  14. 14.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)
  15. 15.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 10404 (b) 9996 (c) 9614 (d) 9604
  16. 16.If a + b = 15 and ab = 36, what is (a − b)²? (a) 81 (b) 225 (c) 9 (d) 153
  17. 17.If a + b = 16 and ab = 55, what is a² + b²? (a) 366 (b) 201 (c) 256 (d) 146
  18. 18.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 4761 (b) 4901 (c) 5041 (d) 5051
  19. 19.If a + b = 14 and ab = 45, what is (a − b)²? (a) 106 (b) 16 (c) 4 (d) 196
  20. 20.What is the product of the roots of x² − 14x + 48 = 0? (a) 48 (b) 47 (c) 49 (d) 14
  21. 21.If a + b = 15 and ab = 50, what is (a − b)²? (a) 225 (b) 125 (c) 5 (d) 25
  22. 22.Which is the correct factorisation of x² + 9x + 18? (a) (x − 3)(x − 6) (b) (x + 1)(x + 8) (c) (x + 3)(x + 6) (d) (x + 1)(x + 18)
  23. 23.If a + b = 17 and ab = 66, what is a² + b²? (a) 157 (b) 223 (c) 289 (d) 421
  24. 24.If a + b = 19 and ab = 84, what is (a − b)²? (a) 193 (b) 361 (c) 5 (d) 25
  25. 25.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 6391 (b) 5929 (c) 6889 (d) 5939
  26. 26.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 896 (b) 1024 (c) 784 (d) 794
  27. 27.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 5625 (b) 4235 (c) 4225 (d) 4875
  28. 28.If a + b = 4 and ab = 3, what is a² + b²? (a) 16 (b) 22 (c) 10 (d) 13
  29. 29.Which is the correct factorisation of x² + 10x + 24? (a) (x + 4)(x + 6) (b) (x + 1)(x + 24) (c) (x − 4)(x − 6) (d) (x + 1)(x + 9)
  30. 30.What is the sum of the roots of x² − 7x + 10 = 0? (a) 10 (b) 7 (c) 6 (d) 8
  31. 31.What is the sum of the roots of x² − 11x + 18 = 0? (a) 18 (b) 12 (c) 11 (d) 10
  32. 32.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 10016 (b) 10826 (c) 10816 (d) 9216
  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 + 40) (d) (x + 1)(x + 12)
  34. 34.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6404 (b) 6724 (c) 6084 (d) 6734
  35. 35.If a + b = 13 and ab = 40, what is a² + b²? (a) 129 (b) 89 (c) 169 (d) 249
  36. 36.Which is the correct factorisation of x² + 9x + 20? (a) (x − 4)(x − 5) (b) (x + 1)(x + 8) (c) (x + 1)(x + 20) (d) (x + 4)(x + 5)
  37. 37.If a + b = 16 and ab = 48, what is (a − b)²? (a) 256 (b) 160 (c) 64 (d) 8
  38. 38.If a + b = 11 and ab = 10, what is (a − b)²? (a) 81 (b) 101 (c) 9 (d) 121
  39. 39.If a + b = 14 and ab = 45, what is a² + b²? (a) 196 (b) 151 (c) 106 (d) 286
  40. 40.If a + b = 7 and ab = 6, what is (a − b)²? (a) 49 (b) 37 (c) 5 (d) 25

Answer key

  1. 30
  2. (x + 3)(x + 7)
  3. 225
  4. 11025
  5. 16
  6. 36
  7. (x + 6)(x + 7)
  8. 441
  9. (x + 4)(x + 8)
  10. 45
  11. 2809
  12. 49
  13. 36
  14. (x + 8)(x + 9)
  15. 9604
  16. 81
  17. 146
  18. 5041
  19. 16
  20. 48
  21. 25
  22. (x + 3)(x + 6)
  23. 157
  24. 25
  25. 5929
  26. 784
  27. 4225
  28. 10
  29. (x + 4)(x + 6)
  30. 7
  31. 11
  32. 10816
  33. (x + 5)(x + 8)
  34. 6724
  35. 89
  36. (x + 4)(x + 5)
  37. 64
  38. 81
  39. 106
  40. 25

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