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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 295 · 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 38²? (a) 1444 (b) 1764 (c) 1454 (d) 1596
  2. 2.What is the sum of the roots of x² − 14x + 48 = 0? (a) 48 (b) 15 (c) 14 (d) 13
  3. 3.What are the roots of x² − 10x + 24 = 0? (a) 4 and 6 (b) 1 and 9 (c) 2 and 12 (d) 1 and 24
  4. 4.If a + b = 19 and ab = 90, what is a² + b²? (a) 181 (b) 271 (c) 361 (d) 541
  5. 5.If a + b = 19 and ab = 84, what is (a − b)²? (a) 193 (b) 5 (c) 25 (d) 361
  6. 6.What are the roots of x² − 7x + 10 = 0? (a) 2 and 5 (b) 1 and 10 (c) 3 and 4 (d) 1 and 6
  7. 7.If a + b = 3 and ab = 2, what is a² + b²? (a) 9 (b) 7 (c) 5 (d) 13
  8. 8.If a + b = 15 and ab = 50, what is a² + b²? (a) 125 (b) 225 (c) 175 (d) 325
  9. 9.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 9) (b) (x − 3)(x − 7) (c) (x + 3)(x + 7) (d) (x + 1)(x + 21)
  10. 10.What is the product of the roots of x² − 7x + 10 = 0? (a) 9 (b) 7 (c) 11 (d) 10
  11. 11.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 6)(x + 8) (c) (x + 1)(x + 13) (d) (x + 1)(x + 48)
  12. 12.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)
  13. 13.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 576 (b) 384 (c) 266 (d) 256
  14. 14.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 16 (c) 14 (d) 15
  15. 15.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 1156 (b) 686 (c) 676 (d) 884
  16. 16.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 8091 (b) 8649 (c) 7579 (d) 7569
  17. 17.If a + b = 6 and ab = 8, what is (a − b)²? (a) 4 (b) 36 (c) 2 (d) 20
  18. 18.If a + b = 13 and ab = 40, what is (a − b)²? (a) 169 (b) 89 (c) 9 (d) 3
  19. 19.If a + b = 13 and ab = 30, what is (a − b)²? (a) 109 (b) 169 (c) 49 (d) 7
  20. 20.Which is the correct factorisation of x² + 9x + 18? (a) (x + 3)(x + 6) (b) (x − 3)(x − 6) (c) (x + 1)(x + 8) (d) (x + 1)(x + 18)
  21. 21.If a + b = 17 and ab = 60, what is (a − b)²? (a) 49 (b) 289 (c) 169 (d) 7
  22. 22.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 9409 (b) 9419 (c) 10609 (d) 9991
  23. 23.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 8) (b) (x − 2)(x − 4) (c) (x + 1)(x + 5) (d) (x + 2)(x + 4)
  24. 24.What is the sum of the roots of x² − 12x + 35 = 0? (a) 13 (b) 11 (c) 35 (d) 12
  25. 25.Which is the correct factorisation of x² + 5x + 6? (a) (x + 2)(x + 3) (b) (x + 1)(x + 4) (c) (x − 2)(x − 3) (d) (x + 1)(x + 6)
  26. 26.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 45) (b) (x + 1)(x + 13) (c) (x − 5)(x − 9) (d) (x + 5)(x + 9)
  27. 27.If a + b = 7 and ab = 10, what is a² + b²? (a) 69 (b) 39 (c) 29 (d) 49
  28. 28.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 6) (b) (x + 1)(x + 12) (c) (x + 3)(x + 4) (d) (x − 3)(x − 4)
  29. 29.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2926 (b) 2516 (c) 2116 (d) 2916
  30. 30.If a + b = 11 and ab = 10, what is a² + b²? (a) 111 (b) 121 (c) 101 (d) 141
  31. 31.Which is the correct factorisation of x² + 12x + 32? (a) (x − 4)(x − 8) (b) (x + 1)(x + 11) (c) (x + 4)(x + 8) (d) (x + 1)(x + 32)
  32. 32.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6571 (b) 6401 (c) 6241 (d) 6561
  33. 33.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 35) (b) (x + 5)(x + 7) (c) (x − 5)(x − 7) (d) (x + 1)(x + 11)
  34. 34.If a + b = 16 and ab = 55, what is a² + b²? (a) 201 (b) 366 (c) 256 (d) 146
  35. 35.What is the sum of the roots of x² − 12x + 32 = 0? (a) 13 (b) 32 (c) 12 (d) 11
  36. 36.If a + b = 14 and ab = 48, what is (a − b)²? (a) 100 (b) 196 (c) 2 (d) 4
  37. 37.If a + b = 15 and ab = 44, what is (a − b)²? (a) 225 (b) 137 (c) 49 (d) 7
  38. 38.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 14) (c) (x + 1)(x + 56) (d) (x + 7)(x + 8)
  39. 39.What is the product of the roots of x² − 14x + 45 = 0? (a) 14 (b) 44 (c) 46 (d) 45
  40. 40.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 6384 (b) 5776 (c) 5786 (d) 7056

Answer key

  1. 1444
  2. 14
  3. 4 and 6
  4. 181
  5. 25
  6. 2 and 5
  7. 5
  8. 125
  9. (x + 3)(x + 7)
  10. 10
  11. (x + 6)(x + 8)
  12. (x + 8)(x + 9)
  13. 256
  14. 15
  15. 676
  16. 7569
  17. 4
  18. 9
  19. 49
  20. (x + 3)(x + 6)
  21. 49
  22. 9409
  23. (x + 2)(x + 4)
  24. 12
  25. (x + 2)(x + 3)
  26. (x + 5)(x + 9)
  27. 29
  28. (x + 3)(x + 4)
  29. 2916
  30. 101
  31. (x + 4)(x + 8)
  32. 6561
  33. (x + 5)(x + 7)
  34. 146
  35. 12
  36. 4
  37. 49
  38. (x + 7)(x + 8)
  39. 45
  40. 5776

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