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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 906 · 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 sum of the roots of x² − 13x + 36 = 0? (a) 14 (b) 12 (c) 36 (d) 13
  2. 2.What is the product of the roots of x² − 12x + 32 = 0? (a) 33 (b) 31 (c) 12 (d) 32
  3. 3.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x + 5)(x + 6) (c) (x − 5)(x − 6) (d) (x + 1)(x + 30)
  4. 4.If a + b = 18 and ab = 80, what is a² + b²? (a) 244 (b) 324 (c) 164 (d) 484
  5. 5.What is the sum of the roots of x² − 7x + 12 = 0? (a) 12 (b) 6 (c) 7 (d) 8
  6. 6.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x − 7)(x − 8) (c) (x + 7)(x + 8) (d) (x + 1)(x + 14)
  7. 7.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8104 (b) 7744 (c) 8474 (d) 8464
  8. 8.What are the roots of x² − 16x + 63 = 0? (a) 1 and 63 (b) 3 and 21 (c) 7 and 9 (d) 1 and 15
  9. 9.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 676 (b) 686 (c) 1156 (d) 884
  10. 10.If a + b = 8 and ab = 15, what is a² + b²? (a) 49 (b) 64 (c) 94 (d) 34
  11. 11.Which is the correct factorisation of x² + 13x + 40? (a) (x − 5)(x − 8) (b) (x + 1)(x + 12) (c) (x + 1)(x + 40) (d) (x + 5)(x + 8)
  12. 12.If a + b = 14 and ab = 24, what is (a − b)²? (a) 196 (b) 10 (c) 148 (d) 100
  13. 13.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)
  14. 14.If a + b = 19 and ab = 88, what is a² + b²? (a) 537 (b) 361 (c) 273 (d) 185
  15. 15.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 13) (b) (x + 1)(x + 45) (c) (x − 5)(x − 9) (d) (x + 5)(x + 9)
  16. 16.What is the product of the roots of x² − 6x + 8 = 0? (a) 9 (b) 8 (c) 6 (d) 7
  17. 17.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 9984 (b) 9216 (c) 9226 (d) 10816
  18. 18.What are the roots of x² − 15x + 54 = 0? (a) 1 and 14 (b) 6 and 9 (c) 2 and 27 (d) 1 and 54
  19. 19.Which is the correct factorisation of x² + 9x + 14? (a) (x − 2)(x − 7) (b) (x + 1)(x + 8) (c) (x + 1)(x + 14) (d) (x + 2)(x + 7)
  20. 20.If a + b = 3 and ab = 2, what is (a − b)²? (a) 1 (b) 5 (c) 7 (d) 9
  21. 21.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 6)(x + 7) (c) (x + 1)(x + 42) (d) (x + 1)(x + 12)
  22. 22.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10004 (b) 10414 (c) 10404 (d) 9604
  23. 23.If a + b = 13 and ab = 30, what is (a − b)²? (a) 169 (b) 7 (c) 109 (d) 49
  24. 24.If a + b = 16 and ab = 55, what is a² + b²? (a) 256 (b) 366 (c) 201 (d) 146
  25. 25.What is the sum of the roots of x² − 11x + 30 = 0? (a) 12 (b) 10 (c) 11 (d) 30
  26. 26.What is the sum of the roots of x² − 5x + 6 = 0? (a) 4 (b) 7 (c) 6 (d) 5
  27. 27.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6399 (b) 6241 (c) 6251 (d) 6561
  28. 28.Which is the correct factorisation of x² + 10x + 24? (a) (x − 4)(x − 6) (b) (x + 4)(x + 6) (c) (x + 1)(x + 24) (d) (x + 1)(x + 9)
  29. 29.Which is the correct factorisation of x² + 10x + 16? (a) (x + 2)(x + 8) (b) (x + 1)(x + 16) (c) (x − 2)(x − 8) (d) (x + 1)(x + 9)
  30. 30.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2926 (b) 2916 (c) 2516 (d) 2116
  31. 31.If a + b = 12 and ab = 20, what is (a − b)²? (a) 104 (b) 144 (c) 64 (d) 8
  32. 32.If a + b = 9 and ab = 20, what is (a − b)²? (a) 81 (b) 1 (c) 61 (d) 41
  33. 33.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 3616 (b) 4106 (c) 4096 (d) 3136
  34. 34.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3731 (b) 3721 (c) 3481 (d) 3601
  35. 35.If a + b = 20 and ab = 96, what is (a − b)²? (a) 400 (b) 208 (c) 4 (d) 16
  36. 36.If a + b = 16 and ab = 60, what is (a − b)²? (a) 256 (b) 16 (c) 4 (d) 136
  37. 37.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 6425 (b) 7235 (c) 7225 (d) 5625
  38. 38.What is the product of the roots of x² − 15x + 56 = 0? (a) 15 (b) 57 (c) 55 (d) 56
  39. 39.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8281 (b) 8291 (c) 8101 (d) 7921
  40. 40.What are the roots of x² − 11x + 30 = 0? (a) 2 and 15 (b) 1 and 10 (c) 1 and 30 (d) 5 and 6

Answer key

  1. 13
  2. 32
  3. (x + 5)(x + 6)
  4. 164
  5. 7
  6. (x + 7)(x + 8)
  7. 8464
  8. 7 and 9
  9. 676
  10. 34
  11. (x + 5)(x + 8)
  12. 100
  13. (x + 8)(x + 9)
  14. 185
  15. (x + 5)(x + 9)
  16. 8
  17. 9216
  18. 6 and 9
  19. (x + 2)(x + 7)
  20. 1
  21. (x + 6)(x + 7)
  22. 10404
  23. 49
  24. 146
  25. 11
  26. 5
  27. 6241
  28. (x + 4)(x + 6)
  29. (x + 2)(x + 8)
  30. 2916
  31. 64
  32. 1
  33. 4096
  34. 3721
  35. 16
  36. 16
  37. 7225
  38. 56
  39. 8281
  40. 5 and 6

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