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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 975 · 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 − 16x + 63 = 0? (a) 1 and 15 (b) 7 and 9 (c) 1 and 63 (d) 3 and 21
  2. 2.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x + 6)(x + 9) (c) (x + 1)(x + 14) (d) (x − 6)(x − 9)
  3. 3.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3364 (b) 3844 (c) 3374 (d) 3596
  4. 4.If a + b = 11 and ab = 30, what is a² + b²? (a) 61 (b) 121 (c) 181 (d) 91
  5. 5.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x + 1)(x + 35) (c) (x + 5)(x + 7) (d) (x − 5)(x − 7)
  6. 6.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 4916 (b) 4356 (c) 5486 (d) 5476
  7. 7.What is the sum of the roots of x² − 16x + 63 = 0? (a) 15 (b) 63 (c) 16 (d) 17
  8. 8.If a + b = 14 and ab = 48, what is a² + b²? (a) 148 (b) 196 (c) 292 (d) 100
  9. 9.If a + b = 17 and ab = 72, what is a² + b²? (a) 289 (b) 217 (c) 145 (d) 433
  10. 10.If a + b = 9 and ab = 18, what is a² + b²? (a) 63 (b) 81 (c) 45 (d) 117
  11. 11.What is the sum of the roots of x² − 12x + 35 = 0? (a) 12 (b) 35 (c) 13 (d) 11
  12. 12.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 851 (b) 899 (c) 961 (d) 841
  13. 13.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 1575 (b) 1235 (c) 1225 (d) 2025
  14. 14.If a + b = 21 and ab = 108, what is (a − b)²? (a) 3 (b) 9 (c) 441 (d) 225
  15. 15.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 5635 (b) 4925 (c) 4225 (d) 5625
  16. 16.If a + b = 8 and ab = 12, what is a² + b²? (a) 40 (b) 52 (c) 64 (d) 88
  17. 17.What is the product of the roots of x² − 7x + 10 = 0? (a) 11 (b) 10 (c) 9 (d) 7
  18. 18.If a + b = 15 and ab = 56, what is a² + b²? (a) 225 (b) 169 (c) 337 (d) 113
  19. 19.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 8) (b) (x + 1)(x + 18) (c) (x + 3)(x + 6) (d) (x − 3)(x − 6)
  20. 20.If a + b = 16 and ab = 63, what is a² + b²? (a) 193 (b) 382 (c) 130 (d) 256
  21. 21.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 73 (c) 72 (d) 17
  22. 22.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x + 6)(x + 7) (c) (x − 6)(x − 7) (d) (x + 1)(x + 42)
  23. 23.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x + 1)(x + 15) (c) (x + 7)(x + 9) (d) (x − 7)(x − 9)
  24. 24.What are the roots of x² − 14x + 48 = 0? (a) 2 and 24 (b) 1 and 13 (c) 6 and 8 (d) 1 and 48
  25. 25.What is the product of the roots of x² − 11x + 18 = 0? (a) 18 (b) 11 (c) 17 (d) 19
  26. 26.What is the product of the roots of x² − 16x + 63 = 0? (a) 62 (b) 16 (c) 64 (d) 63
  27. 27.Which is the correct factorisation of x² + 8x + 15? (a) (x + 3)(x + 5) (b) (x + 1)(x + 15) (c) (x + 1)(x + 7) (d) (x − 3)(x − 5)
  28. 28.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3249 (b) 3979 (c) 3969 (d) 3609
  29. 29.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 1)(x + 13) (c) (x + 6)(x + 8) (d) (x + 1)(x + 48)
  30. 30.Which is the correct factorisation of x² + 9x + 20? (a) (x − 4)(x − 5) (b) (x + 1)(x + 20) (c) (x + 1)(x + 8) (d) (x + 4)(x + 5)
  31. 31.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 8)(x + 9) (c) (x − 8)(x − 9) (d) (x + 1)(x + 72)
  32. 32.If a + b = 20 and ab = 96, what is a² + b²? (a) 592 (b) 400 (c) 208 (d) 304
  33. 33.What is the sum of the roots of x² − 15x + 56 = 0? (a) 16 (b) 14 (c) 56 (d) 15
  34. 34.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 72 (c) 16 (d) 17
  35. 35.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1609 (b) 1369 (c) 1849 (d) 1859
  36. 36.What is the sum of the roots of x² − 13x + 40 = 0? (a) 40 (b) 14 (c) 12 (d) 13
  37. 37.What is the product of the roots of x² − 10x + 24 = 0? (a) 25 (b) 24 (c) 23 (d) 10
  38. 38.If a + b = 20 and ab = 99, what is a² + b²? (a) 400 (b) 598 (c) 202 (d) 301
  39. 39.If a + b = 10 and ab = 21, what is a² + b²? (a) 79 (b) 100 (c) 142 (d) 58
  40. 40.If a + b = 14 and ab = 33, what is (a − b)²? (a) 196 (b) 8 (c) 130 (d) 64

Answer key

  1. 7 and 9
  2. (x + 6)(x + 9)
  3. 3364
  4. 61
  5. (x + 5)(x + 7)
  6. 5476
  7. 16
  8. 100
  9. 145
  10. 45
  11. 12
  12. 841
  13. 1225
  14. 9
  15. 5625
  16. 40
  17. 10
  18. 113
  19. (x + 3)(x + 6)
  20. 130
  21. 72
  22. (x + 6)(x + 7)
  23. (x + 7)(x + 9)
  24. 6 and 8
  25. 18
  26. 63
  27. (x + 3)(x + 5)
  28. 3969
  29. (x + 6)(x + 8)
  30. (x + 4)(x + 5)
  31. (x + 8)(x + 9)
  32. 208
  33. 15
  34. 17
  35. 1849
  36. 13
  37. 24
  38. 202
  39. 58
  40. 64

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