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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 421 · 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.If a + b = 21 and ab = 108, what is (a − b)²? (a) 9 (b) 3 (c) 225 (d) 441
  2. 2.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10619 (b) 9409 (c) 10609 (d) 10009
  3. 3.Which is the correct factorisation of x² + 13x + 36? (a) (x − 4)(x − 9) (b) (x + 1)(x + 36) (c) (x + 4)(x + 9) (d) (x + 1)(x + 12)
  4. 4.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)
  5. 5.If a + b = 19 and ab = 88, what is (a − b)²? (a) 3 (b) 9 (c) 361 (d) 185
  6. 6.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 371 (b) 361 (c) 441 (d) 399
  7. 7.If a + b = 11 and ab = 30, what is a² + b²? (a) 91 (b) 121 (c) 181 (d) 61
  8. 8.What is the sum of the roots of x² − 13x + 42 = 0? (a) 13 (b) 42 (c) 12 (d) 14
  9. 9.If a + b = 15 and ab = 56, what is (a − b)²? (a) 225 (b) 113 (c) 1 (d) 169
  10. 10.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 686 (b) 1156 (c) 884 (d) 676
  11. 11.If a + b = 18 and ab = 80, what is a² + b²? (a) 244 (b) 484 (c) 324 (d) 164
  12. 12.If a + b = 12 and ab = 27, what is (a − b)²? (a) 6 (b) 90 (c) 36 (d) 144
  13. 13.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1296 (b) 1946 (c) 1936 (d) 1616
  14. 14.What is the sum of the roots of x² − 9x + 14 = 0? (a) 14 (b) 10 (c) 8 (d) 9
  15. 15.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 4225 (b) 3035 (c) 3575 (d) 3025
  16. 16.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 384 (b) 256 (c) 266 (d) 576
  17. 17.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 10609 (b) 9409 (c) 9419 (d) 9991
  18. 18.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x + 1)(x + 16) (c) (x − 8)(x − 9) (d) (x + 1)(x + 72)
  19. 19.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x + 1)(x + 8) (c) (x + 4)(x + 5) (d) (x − 4)(x − 5)
  20. 20.If a + b = 20 and ab = 99, what is a² + b²? (a) 598 (b) 400 (c) 301 (d) 202
  21. 21.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9975 (b) 11025 (c) 9035 (d) 9025
  22. 22.If a + b = 12 and ab = 35, what is a² + b²? (a) 144 (b) 214 (c) 74 (d) 109
  23. 23.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6401 (b) 6241 (c) 6561 (d) 6571
  24. 24.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3969 (b) 3591 (c) 3259 (d) 3249
  25. 25.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 3136 (b) 4096 (c) 3146 (d) 3584
  26. 26.If a + b = 14 and ab = 33, what is (a − b)²? (a) 130 (b) 196 (c) 64 (d) 8
  27. 27.If a + b = 19 and ab = 90, what is a² + b²? (a) 541 (b) 271 (c) 361 (d) 181
  28. 28.Which is the correct factorisation of x² + 12x + 27? (a) (x + 3)(x + 9) (b) (x − 3)(x − 9) (c) (x + 1)(x + 27) (d) (x + 1)(x + 11)
  29. 29.What are the roots of x² − 13x + 42 = 0? (a) 6 and 7 (b) 1 and 42 (c) 1 and 12 (d) 2 and 21
  30. 30.Which is the correct factorisation of x² + 8x + 15? (a) (x + 3)(x + 5) (b) (x + 1)(x + 15) (c) (x + 1)(x + 7) (d) (x − 3)(x − 5)
  31. 31.If a + b = 17 and ab = 66, what is a² + b²? (a) 421 (b) 157 (c) 289 (d) 223
  32. 32.What is the product of the roots of x² − 10x + 24 = 0? (a) 23 (b) 24 (c) 25 (d) 10
  33. 33.If a + b = 10 and ab = 21, what is a² + b²? (a) 79 (b) 100 (c) 142 (d) 58
  34. 34.What is the sum of the roots of x² − 7x + 10 = 0? (a) 7 (b) 6 (c) 8 (d) 10
  35. 35.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10414 (b) 10004 (c) 10404 (d) 9604
  36. 36.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 8099 (b) 7931 (c) 8281 (d) 7921
  37. 37.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 1 and 72 (c) 8 and 9 (d) 2 and 36
  38. 38.What is the sum of the roots of x² − 11x + 30 = 0? (a) 10 (b) 30 (c) 12 (d) 11
  39. 39.What is the product of the roots of x² − 11x + 24 = 0? (a) 24 (b) 25 (c) 23 (d) 11
  40. 40.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 901 (b) 841 (c) 971 (d) 961

Answer key

  1. 9
  2. 10609
  3. (x + 4)(x + 9)
  4. (x + 6)(x + 7)
  5. 9
  6. 361
  7. 61
  8. 13
  9. 1
  10. 676
  11. 164
  12. 36
  13. 1936
  14. 9
  15. 3025
  16. 256
  17. 9409
  18. (x + 8)(x + 9)
  19. (x + 4)(x + 5)
  20. 202
  21. 9025
  22. 74
  23. 6561
  24. 3249
  25. 3136
  26. 64
  27. 181
  28. (x + 3)(x + 9)
  29. 6 and 7
  30. (x + 3)(x + 5)
  31. 157
  32. 24
  33. 58
  34. 7
  35. 10404
  36. 7921
  37. 8 and 9
  38. 11
  39. 24
  40. 961

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