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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 198 · 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 product of the roots of x² − 6x + 8 = 0? (a) 7 (b) 9 (c) 8 (d) 6
  2. 2.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x + 5)(x + 6) (c) (x − 5)(x − 6) (d) (x + 1)(x + 30)
  3. 3.What is the product of the roots of x² − 9x + 20 = 0? (a) 9 (b) 20 (c) 21 (d) 19
  4. 4.If a + b = 15 and ab = 36, what is a² + b²? (a) 297 (b) 153 (c) 225 (d) 189
  5. 5.If a + b = 15 and ab = 56, what is (a − b)²? (a) 113 (b) 225 (c) 1 (d) 169
  6. 6.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 10816 (b) 10016 (c) 9216 (d) 10826
  7. 7.Which is the correct factorisation of x² + 9x + 14? (a) (x − 2)(x − 7) (b) (x + 1)(x + 8) (c) (x + 1)(x + 14) (d) (x + 2)(x + 7)
  8. 8.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)
  9. 9.What is the product of the roots of x² − 10x + 21 = 0? (a) 22 (b) 10 (c) 20 (d) 21
  10. 10.What are the roots of x² − 17x + 72 = 0? (a) 8 and 9 (b) 1 and 72 (c) 2 and 36 (d) 1 and 16
  11. 11.If a + b = 19 and ab = 88, what is (a − b)²? (a) 185 (b) 3 (c) 9 (d) 361
  12. 12.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 10) (b) (x − 2)(x − 5) (c) (x + 2)(x + 5) (d) (x + 1)(x + 6)
  13. 13.What is the sum of the roots of x² − 14x + 48 = 0? (a) 15 (b) 14 (c) 48 (d) 13
  14. 14.If a + b = 16 and ab = 63, what is a² + b²? (a) 256 (b) 382 (c) 193 (d) 130
  15. 15.If a + b = 18 and ab = 80, what is (a − b)²? (a) 2 (b) 164 (c) 324 (d) 4
  16. 16.Which is the correct factorisation of x² + 10x + 24? (a) (x − 4)(x − 6) (b) (x + 4)(x + 6) (c) (x + 1)(x + 9) (d) (x + 1)(x + 24)
  17. 17.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 16) (b) (x + 1)(x + 9) (c) (x + 2)(x + 8) (d) (x − 2)(x − 8)
  18. 18.What is the sum of the roots of x² − 15x + 54 = 0? (a) 54 (b) 14 (c) 16 (d) 15
  19. 19.If a + b = 8 and ab = 12, what is a² + b²? (a) 88 (b) 52 (c) 64 (d) 40
  20. 20.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 729 (b) 909 (c) 1089 (d) 1099
  21. 21.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 784 (b) 1024 (c) 904 (d) 1034
  22. 22.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 18 (c) 17 (d) 16
  23. 23.What is the sum of the roots of x² − 10x + 21 = 0? (a) 10 (b) 21 (c) 11 (d) 9
  24. 24.What is the sum of the roots of x² − 7x + 10 = 0? (a) 8 (b) 6 (c) 7 (d) 10
  25. 25.What is the sum of the roots of x² − 13x + 40 = 0? (a) 14 (b) 40 (c) 13 (d) 12
  26. 26.If a + b = 17 and ab = 70, what is a² + b²? (a) 289 (b) 429 (c) 149 (d) 219
  27. 27.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)
  28. 28.If a + b = 16 and ab = 48, what is (a − b)²? (a) 256 (b) 160 (c) 8 (d) 64
  29. 29.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 6) (b) (x − 3)(x − 4) (c) (x + 3)(x + 4) (d) (x + 1)(x + 12)
  30. 30.If a + b = 18 and ab = 80, what is a² + b²? (a) 324 (b) 484 (c) 244 (d) 164
  31. 31.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)
  32. 32.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3721 (b) 3491 (c) 3481 (d) 3599
  33. 33.If a + b = 13 and ab = 30, what is (a − b)²? (a) 109 (b) 49 (c) 169 (d) 7
  34. 34.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10001 (b) 10211 (c) 9801 (d) 10201
  35. 35.If a + b = 17 and ab = 66, what is a² + b²? (a) 223 (b) 289 (c) 421 (d) 157
  36. 36.If a + b = 14 and ab = 48, what is a² + b²? (a) 148 (b) 100 (c) 196 (d) 292
  37. 37.If a + b = 8 and ab = 12, what is (a − b)²? (a) 40 (b) 4 (c) 16 (d) 64
  38. 38.What is the sum of the roots of x² − 16x + 63 = 0? (a) 15 (b) 16 (c) 63 (d) 17
  39. 39.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 1156 (b) 686 (c) 884 (d) 676
  40. 40.What is the sum of the roots of x² − 12x + 35 = 0? (a) 35 (b) 12 (c) 13 (d) 11

Answer key

  1. 8
  2. (x + 5)(x + 6)
  3. 20
  4. 153
  5. 1
  6. 10816
  7. (x + 2)(x + 7)
  8. (x + 6)(x + 7)
  9. 21
  10. 8 and 9
  11. 9
  12. (x + 2)(x + 5)
  13. 14
  14. 130
  15. 4
  16. (x + 4)(x + 6)
  17. (x + 2)(x + 8)
  18. 15
  19. 40
  20. 1089
  21. 1024
  22. 17
  23. 10
  24. 7
  25. 13
  26. 149
  27. (x + 7)(x + 9)
  28. 64
  29. (x + 3)(x + 4)
  30. 164
  31. (x + 8)(x + 9)
  32. 3481
  33. 49
  34. 10201
  35. 157
  36. 100
  37. 16
  38. 16
  39. 676
  40. 12

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