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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 915 · 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 63 (b) 3 and 21 (c) 1 and 15 (d) 7 and 9
  2. 2.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 635 (b) 625 (c) 225 (d) 425
  3. 3.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2304 (b) 2496 (c) 2704 (d) 2314
  4. 4.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 72) (c) (x + 8)(x + 9) (d) (x + 1)(x + 16)
  5. 5.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 1575 (b) 1225 (c) 2025 (d) 1235
  6. 6.If a + b = 16 and ab = 48, what is (a − b)²? (a) 160 (b) 256 (c) 8 (d) 64
  7. 7.If a + b = 19 and ab = 90, what is a² + b²? (a) 181 (b) 271 (c) 361 (d) 541
  8. 8.If a + b = 5 and ab = 6, what is a² + b²? (a) 37 (b) 19 (c) 25 (d) 13
  9. 9.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)
  10. 10.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1859 (b) 1609 (c) 1369 (d) 1849
  11. 11.If a + b = 8 and ab = 12, what is a² + b²? (a) 52 (b) 88 (c) 40 (d) 64
  12. 12.If a + b = 15 and ab = 54, what is (a − b)²? (a) 9 (b) 225 (c) 3 (d) 117
  13. 13.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 16) (b) (x − 2)(x − 8) (c) (x + 2)(x + 8) (d) (x + 1)(x + 9)
  14. 14.If a + b = 12 and ab = 27, what is a² + b²? (a) 90 (b) 198 (c) 117 (d) 144
  15. 15.What are the roots of x² − 10x + 21 = 0? (a) 1 and 21 (b) 1 and 9 (c) 2 and 8 (d) 3 and 7
  16. 16.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 18 (c) 72 (d) 17
  17. 17.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3599 (b) 3491 (c) 3721 (d) 3481
  18. 18.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1521 (b) 1531 (c) 1681 (d) 1599
  19. 19.What is the sum of the roots of x² − 8x + 15 = 0? (a) 15 (b) 8 (c) 7 (d) 9
  20. 20.If a + b = 20 and ab = 96, what is (a − b)²? (a) 4 (b) 400 (c) 16 (d) 208
  21. 21.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)
  22. 22.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 5) (b) (x − 2)(x − 4) (c) (x + 2)(x + 4) (d) (x + 1)(x + 8)
  23. 23.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)
  24. 24.What is the product of the roots of x² − 7x + 10 = 0? (a) 7 (b) 9 (c) 11 (d) 10
  25. 25.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10609 (b) 9409 (c) 10009 (d) 10619
  26. 26.What are the roots of x² − 14x + 45 = 0? (a) 1 and 13 (b) 5 and 9 (c) 3 and 15 (d) 1 and 45
  27. 27.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2926 (b) 2516 (c) 2916 (d) 2116
  28. 28.What is the sum of the roots of x² − 12x + 32 = 0? (a) 12 (b) 11 (c) 32 (d) 13
  29. 29.If a + b = 16 and ab = 60, what is (a − b)²? (a) 16 (b) 136 (c) 256 (d) 4
  30. 30.If a + b = 9 and ab = 18, what is a² + b²? (a) 81 (b) 63 (c) 45 (d) 117
  31. 31.What is the product of the roots of x² − 14x + 48 = 0? (a) 14 (b) 49 (c) 47 (d) 48
  32. 32.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 9801 (b) 10201 (c) 9811 (d) 9999
  33. 33.What are the roots of x² − 12x + 35 = 0? (a) 2 and 10 (b) 1 and 35 (c) 5 and 7 (d) 1 and 11
  34. 34.If a + b = 7 and ab = 10, what is (a − b)²? (a) 49 (b) 29 (c) 3 (d) 9
  35. 35.If a + b = 15 and ab = 54, what is a² + b²? (a) 225 (b) 333 (c) 171 (d) 117
  36. 36.What are the roots of x² − 11x + 18 = 0? (a) 1 and 10 (b) 3 and 6 (c) 1 and 18 (d) 2 and 9
  37. 37.What are the roots of x² − 15x + 56 = 0? (a) 7 and 8 (b) 1 and 56 (c) 1 and 14 (d) 2 and 28
  38. 38.Which is the correct factorisation of x² + 9x + 20? (a) (x − 4)(x − 5) (b) (x + 1)(x + 8) (c) (x + 4)(x + 5) (d) (x + 1)(x + 20)
  39. 39.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 11) (b) (x + 1)(x + 32) (c) (x − 4)(x − 8) (d) (x + 4)(x + 8)
  40. 40.What is the product of the roots of x² − 5x + 6 = 0? (a) 5 (b) 6 (c) 7 (d) 8

Answer key

  1. 7 and 9
  2. 625
  3. 2304
  4. (x + 8)(x + 9)
  5. 1225
  6. 64
  7. 181
  8. 13
  9. (x + 3)(x + 5)
  10. 1849
  11. 40
  12. 9
  13. (x + 2)(x + 8)
  14. 90
  15. 3 and 7
  16. 17
  17. 3481
  18. 1521
  19. 8
  20. 16
  21. (x + 6)(x + 7)
  22. (x + 2)(x + 4)
  23. (x + 2)(x + 3)
  24. 10
  25. 10609
  26. 5 and 9
  27. 2916
  28. 12
  29. 16
  30. 45
  31. 48
  32. 9801
  33. 5 and 7
  34. 9
  35. 117
  36. 2 and 9
  37. 7 and 8
  38. (x + 4)(x + 5)
  39. (x + 4)(x + 8)
  40. 6

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