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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 950 · 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.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2704 (b) 2304 (c) 2314 (d) 2496
  2. 2.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x + 1)(x + 72) (c) (x + 1)(x + 16) (d) (x − 8)(x − 9)
  3. 3.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 841 (b) 961 (c) 971 (d) 901
  4. 4.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 35) (b) (x − 5)(x − 7) (c) (x + 1)(x + 11) (d) (x + 5)(x + 7)
  5. 5.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6241 (b) 6399 (c) 6561 (d) 6251
  6. 6.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2504 (b) 2714 (c) 2704 (d) 2304
  7. 7.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6404 (b) 6084 (c) 6724 (d) 6734
  8. 8.If a + b = 8 and ab = 15, what is (a − b)²? (a) 34 (b) 4 (c) 64 (d) 2
  9. 9.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 3616 (b) 4106 (c) 4096 (d) 3136
  10. 10.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 72 (c) 17 (d) 16
  11. 11.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1609 (b) 1859 (c) 1849 (d) 1369
  12. 12.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 925 (b) 625 (c) 1235 (d) 1225
  13. 13.If a + b = 14 and ab = 33, what is a² + b²? (a) 163 (b) 262 (c) 196 (d) 130
  14. 14.If a + b = 13 and ab = 22, what is (a − b)²? (a) 9 (b) 169 (c) 125 (d) 81
  15. 15.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 635 (b) 225 (c) 425 (d) 625
  16. 16.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 9035 (b) 9025 (c) 8125 (d) 7225
  17. 17.What is the product of the roots of x² − 12x + 32 = 0? (a) 32 (b) 12 (c) 31 (d) 33
  18. 18.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 896 (b) 784 (c) 1024 (d) 794
  19. 19.If a + b = 12 and ab = 35, what is (a − b)²? (a) 2 (b) 74 (c) 144 (d) 4
  20. 20.What is the sum of the roots of x² − 11x + 24 = 0? (a) 10 (b) 11 (c) 24 (d) 12
  21. 21.Which is the correct factorisation of x² + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 1)(x + 12) (c) (x + 1)(x + 6) (d) (x + 3)(x + 4)
  22. 22.If a + b = 4 and ab = 3, what is a² + b²? (a) 10 (b) 22 (c) 16 (d) 13
  23. 23.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x − 7)(x − 9) (c) (x + 1)(x + 63) (d) (x + 7)(x + 9)
  24. 24.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1599 (b) 1681 (c) 1531 (d) 1521
  25. 25.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3364 (b) 3844 (c) 3596 (d) 3374
  26. 26.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 5041 (b) 5051 (c) 4901 (d) 4761
  27. 27.If a + b = 17 and ab = 72, what is (a − b)²? (a) 145 (b) 289 (c) 1 (d) 217
  28. 28.Which is the correct factorisation of x² + 10x + 24? (a) (x + 4)(x + 6) (b) (x − 4)(x − 6) (c) (x + 1)(x + 24) (d) (x + 1)(x + 9)
  29. 29.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 11) (b) (x − 4)(x − 8) (c) (x + 1)(x + 32) (d) (x + 4)(x + 8)
  30. 30.If a + b = 20 and ab = 99, what is (a − b)²? (a) 2 (b) 4 (c) 202 (d) 400
  31. 31.If a + b = 14 and ab = 24, what is a² + b²? (a) 196 (b) 244 (c) 172 (d) 148
  32. 32.What are the roots of x² − 16x + 63 = 0? (a) 1 and 63 (b) 3 and 21 (c) 7 and 9 (d) 1 and 15
  33. 33.If a + b = 9 and ab = 20, what is (a − b)²? (a) 81 (b) 61 (c) 41 (d) 1
  34. 34.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 10816 (b) 10826 (c) 10016 (d) 9216
  35. 35.What is the sum of the roots of x² − 15x + 56 = 0? (a) 15 (b) 56 (c) 14 (d) 16
  36. 36.If a + b = 9 and ab = 18, what is a² + b²? (a) 81 (b) 45 (c) 117 (d) 63
  37. 37.What is the sum of the roots of x² − 14x + 48 = 0? (a) 14 (b) 15 (c) 48 (d) 13
  38. 38.What are the roots of x² − 11x + 28 = 0? (a) 4 and 7 (b) 1 and 10 (c) 2 and 14 (d) 1 and 28
  39. 39.If a + b = 20 and ab = 99, what is a² + b²? (a) 598 (b) 202 (c) 400 (d) 301
  40. 40.Which is the correct factorisation of x² + 14x + 45? (a) (x + 5)(x + 9) (b) (x − 5)(x − 9) (c) (x + 1)(x + 45) (d) (x + 1)(x + 13)

Answer key

  1. 2304
  2. (x + 8)(x + 9)
  3. 961
  4. (x + 5)(x + 7)
  5. 6241
  6. 2704
  7. 6724
  8. 4
  9. 4096
  10. 17
  11. 1849
  12. 1225
  13. 130
  14. 81
  15. 625
  16. 9025
  17. 32
  18. 784
  19. 4
  20. 11
  21. (x + 3)(x + 4)
  22. 10
  23. (x + 7)(x + 9)
  24. 1521
  25. 3364
  26. 5041
  27. 1
  28. (x + 4)(x + 6)
  29. (x + 4)(x + 8)
  30. 4
  31. 148
  32. 7 and 9
  33. 1
  34. 10816
  35. 15
  36. 45
  37. 14
  38. 4 and 7
  39. 202
  40. (x + 5)(x + 9)

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