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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 956 · 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 16²? (a) 266 (b) 576 (c) 384 (d) 256
  2. 2.What is the sum of the roots of x² − 14x + 48 = 0? (a) 48 (b) 15 (c) 13 (d) 14
  3. 3.What is the sum of the roots of x² − 5x + 6 = 0? (a) 4 (b) 6 (c) 5 (d) 7
  4. 4.Which is the correct factorisation of x² + 10x + 16? (a) (x + 2)(x + 8) (b) (x − 2)(x − 8) (c) (x + 1)(x + 16) (d) (x + 1)(x + 9)
  5. 5.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x − 4)(x − 5) (c) (x + 1)(x + 8) (d) (x + 4)(x + 5)
  6. 6.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x + 1)(x + 16) (c) (x + 1)(x + 72) (d) (x − 8)(x − 9)
  7. 7.If a + b = 15 and ab = 44, what is a² + b²? (a) 225 (b) 313 (c) 137 (d) 181
  8. 8.If a + b = 16 and ab = 55, what is (a − b)²? (a) 6 (b) 146 (c) 36 (d) 256
  9. 9.If a + b = 10 and ab = 9, what is (a − b)²? (a) 82 (b) 64 (c) 8 (d) 100
  10. 10.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 10) (b) (x − 3)(x − 8) (c) (x + 1)(x + 24) (d) (x + 3)(x + 8)
  11. 11.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 10404 (b) 9996 (c) 9604 (d) 9614
  12. 12.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 1)(x + 14) (c) (x − 7)(x − 8) (d) (x + 7)(x + 8)
  13. 13.Which is the correct factorisation of x² + 5x + 6? (a) (x + 2)(x + 3) (b) (x + 1)(x + 6) (c) (x − 2)(x − 3) (d) (x + 1)(x + 4)
  14. 14.If a + b = 13 and ab = 12, what is a² + b²? (a) 157 (b) 193 (c) 145 (d) 169
  15. 15.If a + b = 12 and ab = 27, what is a² + b²? (a) 198 (b) 144 (c) 117 (d) 90
  16. 16.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1444 (b) 1454 (c) 1596 (d) 1764
  17. 17.What is the product of the roots of x² − 9x + 18 = 0? (a) 19 (b) 9 (c) 17 (d) 18
  18. 18.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6561 (b) 6571 (c) 6401 (d) 6241
  19. 19.If a + b = 13 and ab = 42, what is (a − b)²? (a) 169 (b) 1 (c) 85 (d) 127
  20. 20.If a + b = 15 and ab = 54, what is a² + b²? (a) 117 (b) 171 (c) 333 (d) 225
  21. 21.What is the sum of the roots of x² − 12x + 35 = 0? (a) 35 (b) 13 (c) 12 (d) 11
  22. 22.If a + b = 14 and ab = 48, what is a² + b²? (a) 100 (b) 196 (c) 292 (d) 148
  23. 23.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 256 (b) 586 (c) 576 (d) 416
  24. 24.Which is the correct factorisation of x² + 9x + 14? (a) (x + 2)(x + 7) (b) (x + 1)(x + 14) (c) (x − 2)(x − 7) (d) (x + 1)(x + 8)
  25. 25.If a + b = 15 and ab = 56, what is (a − b)²? (a) 1 (b) 225 (c) 169 (d) 113
  26. 26.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x − 5)(x − 7) (c) (x + 5)(x + 7) (d) (x + 1)(x + 35)
  27. 27.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 11025 (b) 9025 (c) 11035 (d) 10025
  28. 28.What is the product of the roots of x² − 14x + 45 = 0? (a) 14 (b) 44 (c) 45 (d) 46
  29. 29.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 494 (b) 404 (c) 324 (d) 484
  30. 30.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 18) (b) (x − 3)(x − 6) (c) (x + 3)(x + 6) (d) (x + 1)(x + 8)
  31. 31.Which is the correct factorisation of x² + 12x + 32? (a) (x + 4)(x + 8) (b) (x + 1)(x + 32) (c) (x − 4)(x − 8) (d) (x + 1)(x + 11)
  32. 32.If a + b = 18 and ab = 72, what is a² + b²? (a) 252 (b) 324 (c) 180 (d) 468
  33. 33.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6094 (b) 6396 (c) 6724 (d) 6084
  34. 34.What is the sum of the roots of x² − 13x + 36 = 0? (a) 12 (b) 36 (c) 13 (d) 14
  35. 35.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 1)(x + 63) (c) (x − 7)(x − 9) (d) (x + 7)(x + 9)
  36. 36.If a + b = 19 and ab = 84, what is (a − b)²? (a) 193 (b) 5 (c) 361 (d) 25
  37. 37.What are the roots of x² − 12x + 35 = 0? (a) 1 and 35 (b) 2 and 10 (c) 5 and 7 (d) 1 and 11
  38. 38.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3249 (b) 3979 (c) 3969 (d) 3609
  39. 39.If a + b = 12 and ab = 32, what is (a − b)²? (a) 144 (b) 80 (c) 4 (d) 16
  40. 40.What is the product of the roots of x² − 16x + 63 = 0? (a) 64 (b) 16 (c) 62 (d) 63

Answer key

  1. 256
  2. 14
  3. 5
  4. (x + 2)(x + 8)
  5. (x + 4)(x + 5)
  6. (x + 8)(x + 9)
  7. 137
  8. 36
  9. 64
  10. (x + 3)(x + 8)
  11. 9604
  12. (x + 7)(x + 8)
  13. (x + 2)(x + 3)
  14. 145
  15. 90
  16. 1444
  17. 18
  18. 6561
  19. 1
  20. 117
  21. 12
  22. 100
  23. 576
  24. (x + 2)(x + 7)
  25. 1
  26. (x + 5)(x + 7)
  27. 11025
  28. 45
  29. 484
  30. (x + 3)(x + 6)
  31. (x + 4)(x + 8)
  32. 180
  33. 6084
  34. 13
  35. (x + 7)(x + 9)
  36. 25
  37. 5 and 7
  38. 3969
  39. 16
  40. 63

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