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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 623 · 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 − 11x + 28 = 0? (a) 12 (b) 10 (c) 11 (d) 28
  2. 2.What is the sum of the roots of x² − 9x + 20 = 0? (a) 20 (b) 9 (c) 10 (d) 8
  3. 3.What is the sum of the roots of x² − 13x + 36 = 0? (a) 13 (b) 12 (c) 14 (d) 36
  4. 4.Which is the correct factorisation of x² + 10x + 16? (a) (x − 2)(x − 8) (b) (x + 2)(x + 8) (c) (x + 1)(x + 16) (d) (x + 1)(x + 9)
  5. 5.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6724 (b) 6094 (c) 6396 (d) 6084
  6. 6.If a + b = 17 and ab = 70, what is a² + b²? (a) 289 (b) 149 (c) 219 (d) 429
  7. 7.If a + b = 17 and ab = 60, what is (a − b)²? (a) 7 (b) 289 (c) 169 (d) 49
  8. 8.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 15) (c) (x + 7)(x + 9) (d) (x + 1)(x + 63)
  9. 9.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 8)(x + 9) (c) (x + 1)(x + 72) (d) (x + 1)(x + 16)
  10. 10.If a + b = 17 and ab = 66, what is (a − b)²? (a) 157 (b) 25 (c) 5 (d) 289
  11. 11.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 7921 (b) 8291 (c) 8281 (d) 8101
  12. 12.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 10) (b) (x + 1)(x + 18) (c) (x + 2)(x + 9) (d) (x − 2)(x − 9)
  13. 13.If a + b = 14 and ab = 24, what is (a − b)²? (a) 10 (b) 100 (c) 196 (d) 148
  14. 14.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 1)(x + 13) (c) (x + 1)(x + 48) (d) (x + 6)(x + 8)
  15. 15.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 73 (c) 72 (d) 17
  16. 16.If a + b = 7 and ab = 12, what is a² + b²? (a) 37 (b) 73 (c) 25 (d) 49
  17. 17.If a + b = 12 and ab = 32, what is a² + b²? (a) 112 (b) 208 (c) 80 (d) 144
  18. 18.If a + b = 6 and ab = 8, what is a² + b²? (a) 28 (b) 52 (c) 36 (d) 20
  19. 19.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)
  20. 20.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 4) (b) (x − 2)(x − 3) (c) (x + 2)(x + 3) (d) (x + 1)(x + 6)
  21. 21.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3364 (b) 3596 (c) 3844 (d) 3374
  22. 22.If a + b = 16 and ab = 48, what is (a − b)²? (a) 64 (b) 256 (c) 160 (d) 8
  23. 23.If a + b = 14 and ab = 33, what is a² + b²? (a) 262 (b) 130 (c) 196 (d) 163
  24. 24.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 72 (c) 17 (d) 18
  25. 25.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 5184 (b) 4896 (c) 4624 (d) 4634
  26. 26.What is the product of the roots of x² − 10x + 24 = 0? (a) 25 (b) 23 (c) 24 (d) 10
  27. 27.If a + b = 13 and ab = 22, what is a² + b²? (a) 147 (b) 213 (c) 125 (d) 169
  28. 28.What is the sum of the roots of x² − 8x + 15 = 0? (a) 8 (b) 7 (c) 15 (d) 9
  29. 29.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1369 (b) 1849 (c) 1379 (d) 1591
  30. 30.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 6) (b) (x + 2)(x + 5) (c) (x − 2)(x − 5) (d) (x + 1)(x + 10)
  31. 31.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 539 (b) 529 (c) 289 (d) 409
  32. 32.If a + b = 11 and ab = 10, what is (a − b)²? (a) 101 (b) 121 (c) 81 (d) 9
  33. 33.If a + b = 20 and ab = 99, what is a² + b²? (a) 202 (b) 301 (c) 400 (d) 598
  34. 34.If a + b = 9 and ab = 20, what is (a − b)²? (a) 41 (b) 1 (c) 61 (d) 81
  35. 35.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x − 5)(x − 6) (c) (x + 1)(x + 10) (d) (x + 5)(x + 6)
  36. 36.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 451 (b) 441 (c) 361 (d) 401
  37. 37.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 2 and 36 (c) 1 and 16 (d) 8 and 9
  38. 38.If a + b = 13 and ab = 30, what is a² + b²? (a) 109 (b) 139 (c) 169 (d) 229
  39. 39.If a + b = 18 and ab = 80, what is a² + b²? (a) 164 (b) 324 (c) 244 (d) 484
  40. 40.What are the roots of x² − 14x + 48 = 0? (a) 2 and 24 (b) 1 and 48 (c) 1 and 13 (d) 6 and 8

Answer key

  1. 11
  2. 9
  3. 13
  4. (x + 2)(x + 8)
  5. 6084
  6. 149
  7. 49
  8. (x + 7)(x + 9)
  9. (x + 8)(x + 9)
  10. 25
  11. 8281
  12. (x + 2)(x + 9)
  13. 100
  14. (x + 6)(x + 8)
  15. 72
  16. 25
  17. 80
  18. 20
  19. (x + 5)(x + 7)
  20. (x + 2)(x + 3)
  21. 3364
  22. 64
  23. 130
  24. 17
  25. 4624
  26. 24
  27. 125
  28. 8
  29. 1369
  30. (x + 2)(x + 5)
  31. 529
  32. 81
  33. 202
  34. 1
  35. (x + 5)(x + 6)
  36. 441
  37. 8 and 9
  38. 109
  39. 164
  40. 6 and 8

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