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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 817 · 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 product of the roots of x² − 13x + 42 = 0? (a) 41 (b) 43 (c) 13 (d) 42
  2. 2.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1296 (b) 1616 (c) 1936 (d) 1946
  3. 3.Which is the correct factorisation of x² + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 1)(x + 10) (c) (x + 5)(x + 6) (d) (x + 1)(x + 30)
  4. 4.If a + b = 20 and ab = 99, what is (a − b)²? (a) 2 (b) 400 (c) 202 (d) 4
  5. 5.If a + b = 19 and ab = 84, what is a² + b²? (a) 193 (b) 277 (c) 529 (d) 361
  6. 6.If a + b = 14 and ab = 40, what is a² + b²? (a) 116 (b) 156 (c) 196 (d) 276
  7. 7.What is the sum of the roots of x² − 15x + 56 = 0? (a) 14 (b) 15 (c) 56 (d) 16
  8. 8.What is the sum of the roots of x² − 6x + 8 = 0? (a) 8 (b) 7 (c) 6 (d) 5
  9. 9.What is the sum of the roots of x² − 5x + 6 = 0? (a) 5 (b) 6 (c) 4 (d) 7
  10. 10.If a + b = 15 and ab = 54, what is (a − b)²? (a) 9 (b) 117 (c) 3 (d) 225
  11. 11.What is the sum of the roots of x² − 15x + 54 = 0? (a) 54 (b) 16 (c) 15 (d) 14
  12. 12.If a + b = 19 and ab = 90, what is a² + b²? (a) 271 (b) 181 (c) 361 (d) 541
  13. 13.What are the roots of x² − 16x + 63 = 0? (a) 7 and 9 (b) 1 and 15 (c) 3 and 21 (d) 1 and 63
  14. 14.What is the product of the roots of x² − 13x + 40 = 0? (a) 41 (b) 13 (c) 40 (d) 39
  15. 15.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 7056 (b) 6416 (c) 5776 (d) 7066
  16. 16.What is the product of the roots of x² − 10x + 16 = 0? (a) 10 (b) 16 (c) 17 (d) 15
  17. 17.Which is the correct factorisation of x² + 10x + 24? (a) (x + 4)(x + 6) (b) (x + 1)(x + 9) (c) (x − 4)(x − 6) (d) (x + 1)(x + 24)
  18. 18.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 784 (b) 1024 (c) 1034 (d) 904
  19. 19.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 5194 (b) 4624 (c) 5184 (d) 4904
  20. 20.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 36) (b) (x + 1)(x + 12) (c) (x − 4)(x − 9) (d) (x + 4)(x + 9)
  21. 21.Which is the correct factorisation of x² + 5x + 6? (a) (x − 2)(x − 3) (b) (x + 2)(x + 3) (c) (x + 1)(x + 6) (d) (x + 1)(x + 4)
  22. 22.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 6)(x + 8) (c) (x + 1)(x + 48) (d) (x + 1)(x + 13)
  23. 23.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9996 (b) 10404 (c) 9604 (d) 9614
  24. 24.If a + b = 8 and ab = 12, what is (a − b)²? (a) 4 (b) 40 (c) 64 (d) 16
  25. 25.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 1156 (b) 686 (c) 884 (d) 676
  26. 26.What are the roots of x² − 10x + 16 = 0? (a) 2 and 8 (b) 3 and 7 (c) 1 and 9 (d) 1 and 16
  27. 27.What is the sum of the roots of x² − 13x + 42 = 0? (a) 12 (b) 13 (c) 14 (d) 42
  28. 28.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 15) (b) (x + 3)(x + 5) (c) (x − 3)(x − 5) (d) (x + 1)(x + 7)
  29. 29.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3374 (b) 3364 (c) 3844 (d) 3596
  30. 30.Which is the correct factorisation of x² + 14x + 45? (a) (x + 5)(x + 9) (b) (x − 5)(x − 9) (c) (x + 1)(x + 13) (d) (x + 1)(x + 45)
  31. 31.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 494 (b) 484 (c) 404 (d) 324
  32. 32.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1609 (b) 1369 (c) 1849 (d) 1859
  33. 33.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 15) (c) (x + 7)(x + 9) (d) (x + 1)(x + 63)
  34. 34.What is the product of the roots of x² − 12x + 27 = 0? (a) 12 (b) 28 (c) 27 (d) 26
  35. 35.If a + b = 5 and ab = 4, what is a² + b²? (a) 33 (b) 21 (c) 25 (d) 17
  36. 36.What is the product of the roots of x² − 10x + 24 = 0? (a) 10 (b) 23 (c) 24 (d) 25
  37. 37.If a + b = 5 and ab = 6, what is (a − b)²? (a) 1 (b) 13 (c) 19 (d) 25
  38. 38.What is the sum of the roots of x² − 13x + 36 = 0? (a) 12 (b) 36 (c) 14 (d) 13
  39. 39.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 71 (c) 72 (d) 73
  40. 40.If a + b = 19 and ab = 88, what is (a − b)²? (a) 361 (b) 9 (c) 3 (d) 185

Answer key

  1. 42
  2. 1936
  3. (x + 5)(x + 6)
  4. 4
  5. 193
  6. 116
  7. 15
  8. 6
  9. 5
  10. 9
  11. 15
  12. 181
  13. 7 and 9
  14. 40
  15. 7056
  16. 16
  17. (x + 4)(x + 6)
  18. 1024
  19. 5184
  20. (x + 4)(x + 9)
  21. (x + 2)(x + 3)
  22. (x + 6)(x + 8)
  23. 9604
  24. 16
  25. 676
  26. 2 and 8
  27. 13
  28. (x + 3)(x + 5)
  29. 3364
  30. (x + 5)(x + 9)
  31. 484
  32. 1849
  33. (x + 7)(x + 9)
  34. 27
  35. 17
  36. 24
  37. 1
  38. 13
  39. 72
  40. 9

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