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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 259 · 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.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x + 1)(x + 10) (c) (x + 5)(x + 6) (d) (x − 5)(x − 6)
  2. 2.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 8096 (b) 7754 (c) 7744 (d) 8464
  3. 3.If a + b = 7 and ab = 12, what is (a − b)²? (a) 49 (b) 1 (c) 25 (d) 37
  4. 4.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 10816 (b) 9216 (c) 10016 (d) 10826
  5. 5.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6084 (b) 6094 (c) 6724 (d) 6396
  6. 6.What is the product of the roots of x² − 12x + 32 = 0? (a) 31 (b) 12 (c) 32 (d) 33
  7. 7.If a + b = 7 and ab = 10, what is (a − b)²? (a) 49 (b) 29 (c) 9 (d) 3
  8. 8.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 72 (c) 18 (d) 17
  9. 9.If a + b = 17 and ab = 66, what is (a − b)²? (a) 289 (b) 5 (c) 157 (d) 25
  10. 10.If a + b = 17 and ab = 72, what is a² + b²? (a) 217 (b) 145 (c) 433 (d) 289
  11. 11.If a + b = 10 and ab = 9, what is (a − b)²? (a) 82 (b) 64 (c) 100 (d) 8
  12. 12.If a + b = 17 and ab = 66, what is a² + b²? (a) 223 (b) 157 (c) 289 (d) 421
  13. 13.Which is the correct factorisation of x² + 9x + 14? (a) (x − 2)(x − 7) (b) (x + 1)(x + 14) (c) (x + 2)(x + 7) (d) (x + 1)(x + 8)
  14. 14.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 371 (b) 361 (c) 441 (d) 399
  15. 15.If a + b = 11 and ab = 30, what is (a − b)²? (a) 121 (b) 1 (c) 61 (d) 91
  16. 16.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 16) (c) (x + 8)(x + 9) (d) (x + 1)(x + 72)
  17. 17.If a + b = 20 and ab = 96, what is a² + b²? (a) 208 (b) 400 (c) 592 (d) 304
  18. 18.Which is the correct factorisation of x² + 12x + 32? (a) (x − 4)(x − 8) (b) (x + 1)(x + 11) (c) (x + 4)(x + 8) (d) (x + 1)(x + 32)
  19. 19.What is the product of the roots of x² − 9x + 18 = 0? (a) 17 (b) 18 (c) 19 (d) 9
  20. 20.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 8125 (b) 7225 (c) 9035 (d) 9025
  21. 21.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2516 (b) 2916 (c) 2926 (d) 2116
  22. 22.What are the roots of x² − 7x + 10 = 0? (a) 1 and 10 (b) 3 and 4 (c) 1 and 6 (d) 2 and 5
  23. 23.What is the sum of the roots of x² − 11x + 30 = 0? (a) 30 (b) 10 (c) 11 (d) 12
  24. 24.What is the product of the roots of x² − 11x + 30 = 0? (a) 11 (b) 31 (c) 30 (d) 29
  25. 25.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x + 1)(x + 12) (c) (x − 6)(x − 7) (d) (x + 6)(x + 7)
  26. 26.What is the sum of the roots of x² − 11x + 24 = 0? (a) 10 (b) 11 (c) 24 (d) 12
  27. 27.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2601 (b) 2501 (c) 2401 (d) 2611
  28. 28.If a + b = 8 and ab = 7, what is a² + b²? (a) 64 (b) 57 (c) 78 (d) 50
  29. 29.If a + b = 12 and ab = 32, what is a² + b²? (a) 208 (b) 80 (c) 144 (d) 112
  30. 30.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 9801 (b) 10211 (c) 10201 (d) 10001
  31. 31.If a + b = 16 and ab = 55, what is a² + b²? (a) 256 (b) 201 (c) 146 (d) 366
  32. 32.If a + b = 11 and ab = 18, what is (a − b)²? (a) 49 (b) 7 (c) 85 (d) 121
  33. 33.What are the roots of x² − 8x + 12 = 0? (a) 1 and 7 (b) 3 and 4 (c) 1 and 12 (d) 2 and 6
  34. 34.What are the roots of x² − 11x + 28 = 0? (a) 2 and 14 (b) 1 and 10 (c) 1 and 28 (d) 4 and 7
  35. 35.What is the sum of the roots of x² − 16x + 63 = 0? (a) 16 (b) 15 (c) 63 (d) 17
  36. 36.What is the sum of the roots of x² − 12x + 35 = 0? (a) 13 (b) 11 (c) 35 (d) 12
  37. 37.If a + b = 8 and ab = 15, what is (a − b)²? (a) 2 (b) 64 (c) 4 (d) 34
  38. 38.If a + b = 13 and ab = 40, what is (a − b)²? (a) 89 (b) 169 (c) 9 (d) 3
  39. 39.If a + b = 17 and ab = 70, what is (a − b)²? (a) 9 (b) 149 (c) 289 (d) 3
  40. 40.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8291 (b) 7921 (c) 8281 (d) 8101

Answer key

  1. (x + 5)(x + 6)
  2. 7744
  3. 1
  4. 10816
  5. 6084
  6. 32
  7. 9
  8. 17
  9. 25
  10. 145
  11. 64
  12. 157
  13. (x + 2)(x + 7)
  14. 361
  15. 1
  16. (x + 8)(x + 9)
  17. 208
  18. (x + 4)(x + 8)
  19. 18
  20. 9025
  21. 2916
  22. 2 and 5
  23. 11
  24. 30
  25. (x + 6)(x + 7)
  26. 11
  27. 2601
  28. 50
  29. 80
  30. 10201
  31. 146
  32. 49
  33. 2 and 6
  34. 4 and 7
  35. 16
  36. 12
  37. 4
  38. 9
  39. 9
  40. 8281

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