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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 488 · 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 − 8x + 12 = 0? (a) 7 (b) 8 (c) 9 (d) 12
  2. 2.What is the sum of the roots of x² − 7x + 10 = 0? (a) 7 (b) 6 (c) 8 (d) 10
  3. 3.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9604 (b) 10404 (c) 9996 (d) 9614
  4. 4.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 7931 (b) 8099 (c) 8281 (d) 7921
  5. 5.What is the product of the roots of x² − 11x + 24 = 0? (a) 25 (b) 11 (c) 23 (d) 24
  6. 6.What are the roots of x² − 12x + 32 = 0? (a) 4 and 8 (b) 1 and 11 (c) 1 and 32 (d) 2 and 16
  7. 7.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 12) (b) (x + 3)(x + 4) (c) (x − 3)(x − 4) (d) (x + 1)(x + 6)
  8. 8.If a + b = 20 and ab = 96, what is a² + b²? (a) 400 (b) 592 (c) 304 (d) 208
  9. 9.What is the product of the roots of x² − 9x + 14 = 0? (a) 13 (b) 9 (c) 14 (d) 15
  10. 10.If a + b = 9 and ab = 18, what is a² + b²? (a) 45 (b) 63 (c) 81 (d) 117
  11. 11.If a + b = 13 and ab = 40, what is a² + b²? (a) 89 (b) 129 (c) 169 (d) 249
  12. 12.If a + b = 21 and ab = 108, what is a² + b²? (a) 657 (b) 333 (c) 441 (d) 225
  13. 13.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 2525 (b) 2025 (c) 3035 (d) 3025
  14. 14.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)
  15. 15.What are the roots of x² − 13x + 40 = 0? (a) 1 and 12 (b) 1 and 40 (c) 5 and 8 (d) 2 and 20
  16. 16.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 15) (c) (x + 1)(x + 63) (d) (x + 7)(x + 9)
  17. 17.What is the sum of the roots of x² − 10x + 24 = 0? (a) 24 (b) 10 (c) 11 (d) 9
  18. 18.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 399 (b) 371 (c) 441 (d) 361
  19. 19.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 18 (c) 16 (d) 17
  20. 20.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 7579 (b) 8091 (c) 7569 (d) 8649
  21. 21.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x − 7)(x − 8) (c) (x + 1)(x + 14) (d) (x + 7)(x + 8)
  22. 22.Which is the correct factorisation of x² + 7x + 10? (a) (x + 2)(x + 5) (b) (x − 2)(x − 5) (c) (x + 1)(x + 10) (d) (x + 1)(x + 6)
  23. 23.If a + b = 9 and ab = 14, what is a² + b²? (a) 109 (b) 53 (c) 67 (d) 81
  24. 24.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 586 (b) 576 (c) 416 (d) 256
  25. 25.If a + b = 18 and ab = 72, what is a² + b²? (a) 324 (b) 252 (c) 180 (d) 468
  26. 26.What is the product of the roots of x² − 13x + 40 = 0? (a) 39 (b) 41 (c) 40 (d) 13
  27. 27.If a + b = 15 and ab = 54, what is (a − b)²? (a) 225 (b) 3 (c) 117 (d) 9
  28. 28.What are the roots of x² − 9x + 18 = 0? (a) 3 and 6 (b) 1 and 8 (c) 2 and 9 (d) 1 and 18
  29. 29.If a + b = 19 and ab = 84, what is (a − b)²? (a) 5 (b) 193 (c) 25 (d) 361
  30. 30.If a + b = 7 and ab = 6, what is (a − b)²? (a) 37 (b) 5 (c) 49 (d) 25
  31. 31.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 1)(x + 12) (c) (x + 6)(x + 7) (d) (x + 1)(x + 42)
  32. 32.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1454 (b) 1764 (c) 1444 (d) 1596
  33. 33.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2516 (b) 2116 (c) 2926 (d) 2916
  34. 34.If a + b = 8 and ab = 12, what is a² + b²? (a) 88 (b) 52 (c) 64 (d) 40
  35. 35.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1369 (b) 1609 (c) 1849 (d) 1859
  36. 36.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)
  37. 37.If a + b = 8 and ab = 15, what is (a − b)²? (a) 4 (b) 64 (c) 2 (d) 34
  38. 38.What is the sum of the roots of x² − 6x + 8 = 0? (a) 8 (b) 7 (c) 6 (d) 5
  39. 39.If a + b = 17 and ab = 72, what is (a − b)²? (a) 217 (b) 289 (c) 1 (d) 145
  40. 40.What is the product of the roots of x² − 13x + 42 = 0? (a) 43 (b) 41 (c) 13 (d) 42

Answer key

  1. 8
  2. 7
  3. 9604
  4. 7921
  5. 24
  6. 4 and 8
  7. (x + 3)(x + 4)
  8. 208
  9. 14
  10. 45
  11. 89
  12. 225
  13. 3025
  14. (x + 8)(x + 9)
  15. 5 and 8
  16. (x + 7)(x + 9)
  17. 10
  18. 361
  19. 17
  20. 7569
  21. (x + 7)(x + 8)
  22. (x + 2)(x + 5)
  23. 53
  24. 576
  25. 180
  26. 40
  27. 9
  28. 3 and 6
  29. 25
  30. 25
  31. (x + 6)(x + 7)
  32. 1444
  33. 2916
  34. 40
  35. 1849
  36. (x + 4)(x + 9)
  37. 4
  38. 6
  39. 1
  40. 42

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