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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 312 · 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 are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 1 and 72 (c) 8 and 9 (d) 2 and 36
  2. 2.If a + b = 8 and ab = 15, what is a² + b²? (a) 34 (b) 64 (c) 94 (d) 49
  3. 3.If a + b = 7 and ab = 12, what is a² + b²? (a) 73 (b) 25 (c) 37 (d) 49
  4. 4.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 71 (c) 72 (d) 73
  5. 5.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 635 (b) 1225 (c) 875 (d) 625
  6. 6.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2501 (b) 2611 (c) 2401 (d) 2601
  7. 7.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x − 5)(x − 6) (c) (x + 1)(x + 10) (d) (x + 5)(x + 6)
  8. 8.What is the sum of the roots of x² − 13x + 42 = 0? (a) 42 (b) 13 (c) 14 (d) 12
  9. 9.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)
  10. 10.If a + b = 3 and ab = 2, what is (a − b)²? (a) 7 (b) 5 (c) 9 (d) 1
  11. 11.If a + b = 9 and ab = 8, what is a² + b²? (a) 97 (b) 81 (c) 73 (d) 65
  12. 12.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9996 (b) 9614 (c) 9604 (d) 10404
  13. 13.If a + b = 9 and ab = 20, what is (a − b)²? (a) 41 (b) 1 (c) 61 (d) 81
  14. 14.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 16 (c) 72 (d) 17
  15. 15.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3374 (b) 3596 (c) 3364 (d) 3844
  16. 16.Which is the correct factorisation of x² + 10x + 21? (a) (x − 3)(x − 7) (b) (x + 3)(x + 7) (c) (x + 1)(x + 21) (d) (x + 1)(x + 9)
  17. 17.What is the sum of the roots of x² − 10x + 16 = 0? (a) 9 (b) 11 (c) 10 (d) 16
  18. 18.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1369 (b) 1609 (c) 1849 (d) 1859
  19. 19.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 11) (b) (x + 3)(x + 9) (c) (x − 3)(x − 9) (d) (x + 1)(x + 27)
  20. 20.If a + b = 6 and ab = 5, what is a² + b²? (a) 36 (b) 46 (c) 31 (d) 26
  21. 21.What is the product of the roots of x² − 8x + 12 = 0? (a) 12 (b) 13 (c) 11 (d) 8
  22. 22.If a + b = 7 and ab = 6, what is a² + b²? (a) 37 (b) 43 (c) 61 (d) 49
  23. 23.What is the sum of the roots of x² − 9x + 14 = 0? (a) 10 (b) 14 (c) 8 (d) 9
  24. 24.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 9) (b) (x − 4)(x − 6) (c) (x + 1)(x + 24) (d) (x + 4)(x + 6)
  25. 25.If a + b = 16 and ab = 60, what is (a − b)²? (a) 16 (b) 4 (c) 256 (d) 136
  26. 26.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x − 5)(x − 7) (c) (x + 1)(x + 35) (d) (x + 5)(x + 7)
  27. 27.If a + b = 13 and ab = 42, what is (a − b)²? (a) 127 (b) 169 (c) 1 (d) 85
  28. 28.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 784 (b) 896 (c) 794 (d) 1024
  29. 29.If a + b = 21 and ab = 108, what is a² + b²? (a) 441 (b) 333 (c) 657 (d) 225
  30. 30.What is the sum of the roots of x² − 15x + 56 = 0? (a) 14 (b) 56 (c) 16 (d) 15
  31. 31.If a + b = 11 and ab = 24, what is a² + b²? (a) 97 (b) 73 (c) 121 (d) 169
  32. 32.If a + b = 13 and ab = 12, what is a² + b²? (a) 193 (b) 145 (c) 157 (d) 169
  33. 33.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 10) (b) (x − 2)(x − 5) (c) (x + 1)(x + 6) (d) (x + 2)(x + 5)
  34. 34.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2126 (b) 2484 (c) 2116 (d) 2916
  35. 35.If a + b = 13 and ab = 40, what is (a − b)²? (a) 3 (b) 89 (c) 9 (d) 169
  36. 36.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 7921 (b) 8101 (c) 8281 (d) 8291
  37. 37.If a + b = 17 and ab = 66, what is (a − b)²? (a) 5 (b) 25 (c) 157 (d) 289
  38. 38.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10404 (b) 10004 (c) 10414 (d) 9604
  39. 39.If a + b = 19 and ab = 90, what is a² + b²? (a) 361 (b) 541 (c) 181 (d) 271
  40. 40.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 256 (b) 576 (c) 416 (d) 586

Answer key

  1. 8 and 9
  2. 34
  3. 25
  4. 72
  5. 625
  6. 2601
  7. (x + 5)(x + 6)
  8. 13
  9. (x + 8)(x + 9)
  10. 1
  11. 65
  12. 9604
  13. 1
  14. 17
  15. 3364
  16. (x + 3)(x + 7)
  17. 10
  18. 1849
  19. (x + 3)(x + 9)
  20. 26
  21. 12
  22. 37
  23. 9
  24. (x + 4)(x + 6)
  25. 16
  26. (x + 5)(x + 7)
  27. 1
  28. 784
  29. 225
  30. 15
  31. 73
  32. 145
  33. (x + 2)(x + 5)
  34. 2116
  35. 9
  36. 8281
  37. 25
  38. 10404
  39. 181
  40. 576

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