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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 15 · 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 + 8)(x + 9) (b) (x + 1)(x + 72) (c) (x − 8)(x − 9) (d) (x + 1)(x + 16)
  2. 2.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8101 (b) 8281 (c) 7921 (d) 8291
  3. 3.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 9409 (b) 10619 (c) 10609 (d) 10009
  4. 4.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 576 (b) 266 (c) 384 (d) 256
  5. 5.If a + b = 5 and ab = 4, what is a² + b²? (a) 17 (b) 21 (c) 33 (d) 25
  6. 6.If a + b = 10 and ab = 24, what is (a − b)²? (a) 52 (b) 100 (c) 4 (d) 2
  7. 7.What is the sum of the roots of x² − 15x + 54 = 0? (a) 16 (b) 54 (c) 15 (d) 14
  8. 8.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)
  9. 9.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2516 (b) 2116 (c) 2926 (d) 2916
  10. 10.If a + b = 14 and ab = 40, what is a² + b²? (a) 196 (b) 116 (c) 276 (d) 156
  11. 11.What are the roots of x² − 10x + 16 = 0? (a) 3 and 7 (b) 2 and 8 (c) 1 and 16 (d) 1 and 9
  12. 12.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 916 (b) 676 (c) 1166 (d) 1156
  13. 13.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10004 (b) 10404 (c) 9604 (d) 10414
  14. 14.If a + b = 19 and ab = 88, what is (a − b)²? (a) 3 (b) 9 (c) 361 (d) 185
  15. 15.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 6899 (b) 6889 (c) 5929 (d) 6409
  16. 16.If a + b = 19 and ab = 90, what is (a − b)²? (a) 181 (b) 271 (c) 361 (d) 1
  17. 17.Which is the correct factorisation of x² + 8x + 15? (a) (x + 3)(x + 5) (b) (x + 1)(x + 7) (c) (x + 1)(x + 15) (d) (x − 3)(x − 5)
  18. 18.Which is the correct factorisation of x² + 11x + 18? (a) (x − 2)(x − 9) (b) (x + 1)(x + 10) (c) (x + 1)(x + 18) (d) (x + 2)(x + 9)
  19. 19.Which is the correct factorisation of x² + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 1)(x + 30) (c) (x + 1)(x + 10) (d) (x + 5)(x + 6)
  20. 20.If a + b = 14 and ab = 48, what is a² + b²? (a) 196 (b) 292 (c) 148 (d) 100
  21. 21.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 3136 (b) 4096 (c) 4106 (d) 3616
  22. 22.If a + b = 3 and ab = 2, what is (a − b)²? (a) 5 (b) 7 (c) 9 (d) 1
  23. 23.If a + b = 7 and ab = 12, what is (a − b)²? (a) 25 (b) 1 (c) 49 (d) 37
  24. 24.What is the sum of the roots of x² − 8x + 15 = 0? (a) 15 (b) 8 (c) 9 (d) 7
  25. 25.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4761 (b) 4899 (c) 4771 (d) 5041
  26. 26.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 56) (c) (x + 7)(x + 8) (d) (x + 1)(x + 14)
  27. 27.What are the roots of x² − 7x + 12 = 0? (a) 1 and 6 (b) 1 and 12 (c) 2 and 6 (d) 3 and 4
  28. 28.If a + b = 13 and ab = 22, what is a² + b²? (a) 213 (b) 125 (c) 147 (d) 169
  29. 29.If a + b = 17 and ab = 70, what is a² + b²? (a) 219 (b) 289 (c) 429 (d) 149
  30. 30.If a + b = 16 and ab = 60, what is a² + b²? (a) 256 (b) 376 (c) 196 (d) 136
  31. 31.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10201 (b) 9801 (c) 10211 (d) 10001
  32. 32.If a + b = 12 and ab = 11, what is a² + b²? (a) 144 (b) 133 (c) 122 (d) 166
  33. 33.Which is the correct factorisation of x² + 13x + 36? (a) (x + 4)(x + 9) (b) (x + 1)(x + 12) (c) (x − 4)(x − 9) (d) (x + 1)(x + 36)
  34. 34.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 24) (b) (x − 3)(x − 8) (c) (x + 3)(x + 8) (d) (x + 1)(x + 10)
  35. 35.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 15 (c) 14 (d) 16
  36. 36.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3481 (b) 3721 (c) 3601 (d) 3731
  37. 37.What are the roots of x² − 12x + 35 = 0? (a) 2 and 10 (b) 1 and 11 (c) 5 and 7 (d) 1 and 35
  38. 38.If a + b = 6 and ab = 8, what is a² + b²? (a) 20 (b) 28 (c) 52 (d) 36
  39. 39.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 7931 (b) 8099 (c) 8281 (d) 7921
  40. 40.What is the product of the roots of x² − 16x + 63 = 0? (a) 62 (b) 16 (c) 64 (d) 63

Answer key

  1. (x + 8)(x + 9)
  2. 8281
  3. 10609
  4. 256
  5. 17
  6. 4
  7. 15
  8. (x + 2)(x + 7)
  9. 2916
  10. 116
  11. 2 and 8
  12. 1156
  13. 10404
  14. 9
  15. 6889
  16. 1
  17. (x + 3)(x + 5)
  18. (x + 2)(x + 9)
  19. (x + 5)(x + 6)
  20. 100
  21. 4096
  22. 1
  23. 1
  24. 8
  25. 4761
  26. (x + 7)(x + 8)
  27. 3 and 4
  28. 125
  29. 149
  30. 136
  31. 10201
  32. 122
  33. (x + 4)(x + 9)
  34. (x + 3)(x + 8)
  35. 15
  36. 3721
  37. 5 and 7
  38. 20
  39. 7921
  40. 63

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