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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 574 · 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 − 10x + 21 = 0? (a) 11 (b) 21 (c) 10 (d) 9
  2. 2.If a + b = 18 and ab = 72, what is (a − b)²? (a) 324 (b) 6 (c) 36 (d) 180
  3. 3.If a + b = 15 and ab = 36, what is (a − b)²? (a) 9 (b) 225 (c) 153 (d) 81
  4. 4.If a + b = 15 and ab = 54, what is a² + b²? (a) 117 (b) 333 (c) 225 (d) 171
  5. 5.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 2025 (b) 3025 (c) 2525 (d) 3035
  6. 6.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 8) (b) (x + 4)(x + 5) (c) (x + 1)(x + 20) (d) (x − 4)(x − 5)
  7. 7.If a + b = 20 and ab = 99, what is (a − b)²? (a) 2 (b) 400 (c) 202 (d) 4
  8. 8.If a + b = 12 and ab = 27, what is a² + b²? (a) 198 (b) 117 (c) 144 (d) 90
  9. 9.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 1)(x + 12) (c) (x + 1)(x + 42) (d) (x + 6)(x + 7)
  10. 10.If a + b = 7 and ab = 10, what is (a − b)²? (a) 29 (b) 49 (c) 3 (d) 9
  11. 11.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 6) (b) (x + 1)(x + 10) (c) (x + 2)(x + 5) (d) (x − 2)(x − 5)
  12. 12.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 1225 (b) 625 (c) 635 (d) 875
  13. 13.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 5339 (b) 4489 (c) 5329 (d) 4909
  14. 14.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 1166 (b) 676 (c) 916 (d) 1156
  15. 15.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 9811 (b) 10201 (c) 9801 (d) 9999
  16. 16.If a + b = 13 and ab = 30, what is a² + b²? (a) 229 (b) 109 (c) 169 (d) 139
  17. 17.If a + b = 9 and ab = 20, what is (a − b)²? (a) 41 (b) 61 (c) 1 (d) 81
  18. 18.If a + b = 12 and ab = 35, what is (a − b)²? (a) 74 (b) 2 (c) 144 (d) 4
  19. 19.If a + b = 17 and ab = 70, what is (a − b)²? (a) 3 (b) 289 (c) 9 (d) 149
  20. 20.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x + 1)(x + 15) (c) (x + 1)(x + 63) (d) (x − 7)(x − 9)
  21. 21.If a + b = 9 and ab = 18, what is a² + b²? (a) 81 (b) 45 (c) 117 (d) 63
  22. 22.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x − 6)(x − 8) (c) (x + 6)(x + 8) (d) (x + 1)(x + 13)
  23. 23.If a + b = 21 and ab = 108, what is a² + b²? (a) 333 (b) 441 (c) 657 (d) 225
  24. 24.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 14) (b) (x + 1)(x + 54) (c) (x − 6)(x − 9) (d) (x + 6)(x + 9)
  25. 25.If a + b = 14 and ab = 45, what is (a − b)²? (a) 4 (b) 106 (c) 16 (d) 196
  26. 26.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 391 (b) 299 (c) 289 (d) 529
  27. 27.If a + b = 17 and ab = 66, what is (a − b)²? (a) 5 (b) 157 (c) 25 (d) 289
  28. 28.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 784 (b) 904 (c) 1034 (d) 1024
  29. 29.If a + b = 18 and ab = 80, what is a² + b²? (a) 164 (b) 484 (c) 244 (d) 324
  30. 30.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 4624 (b) 5184 (c) 4634 (d) 4896
  31. 31.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3596 (b) 3844 (c) 3364 (d) 3374
  32. 32.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 18 (c) 16 (d) 17
  33. 33.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6561 (b) 6241 (c) 6251 (d) 6399
  34. 34.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1691 (b) 1601 (c) 1521 (d) 1681
  35. 35.If a + b = 15 and ab = 56, what is (a − b)²? (a) 225 (b) 169 (c) 113 (d) 1
  36. 36.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2496 (b) 2304 (c) 2314 (d) 2704
  37. 37.What is the sum of the roots of x² − 12x + 32 = 0? (a) 11 (b) 32 (c) 13 (d) 12
  38. 38.If a + b = 5 and ab = 6, what is (a − b)²? (a) 25 (b) 13 (c) 1 (d) 19
  39. 39.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10211 (b) 10201 (c) 9801 (d) 10001
  40. 40.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

Answer key

  1. 10
  2. 36
  3. 81
  4. 117
  5. 3025
  6. (x + 4)(x + 5)
  7. 4
  8. 90
  9. (x + 6)(x + 7)
  10. 9
  11. (x + 2)(x + 5)
  12. 625
  13. 5329
  14. 1156
  15. 9801
  16. 109
  17. 1
  18. 4
  19. 9
  20. (x + 7)(x + 9)
  21. 45
  22. (x + 6)(x + 8)
  23. 225
  24. (x + 6)(x + 9)
  25. 16
  26. 289
  27. 25
  28. 1024
  29. 164
  30. 4624
  31. 3364
  32. 17
  33. 6241
  34. 1681
  35. 1
  36. 2304
  37. 12
  38. 1
  39. 10201
  40. 2 and 6

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