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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 275 · 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 + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 56) (c) (x + 7)(x + 8) (d) (x + 1)(x + 14)
  2. 2.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x − 8)(x − 9) (c) (x + 1)(x + 16) (d) (x + 1)(x + 72)
  3. 3.If a + b = 16 and ab = 48, what is a² + b²? (a) 256 (b) 208 (c) 352 (d) 160
  4. 4.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2496 (b) 2304 (c) 2314 (d) 2704
  5. 5.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 5) (b) (x + 2)(x + 4) (c) (x − 2)(x − 4) (d) (x + 1)(x + 8)
  6. 6.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1849 (b) 1379 (c) 1369 (d) 1591
  7. 7.If a + b = 20 and ab = 99, what is a² + b²? (a) 400 (b) 301 (c) 598 (d) 202
  8. 8.Which is the correct factorisation of x² + 12x + 27? (a) (x − 3)(x − 9) (b) (x + 3)(x + 9) (c) (x + 1)(x + 11) (d) (x + 1)(x + 27)
  9. 9.If a + b = 13 and ab = 42, what is a² + b²? (a) 127 (b) 169 (c) 85 (d) 253
  10. 10.If a + b = 12 and ab = 35, what is a² + b²? (a) 74 (b) 214 (c) 144 (d) 109
  11. 11.If a + b = 18 and ab = 77, what is (a − b)²? (a) 16 (b) 4 (c) 170 (d) 324
  12. 12.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 361 (b) 451 (c) 441 (d) 401
  13. 13.What are the roots of x² − 8x + 15 = 0? (a) 1 and 15 (b) 3 and 5 (c) 1 and 7 (d) 2 and 6
  14. 14.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 16 (c) 72 (d) 17
  15. 15.If a + b = 17 and ab = 72, what is a² + b²? (a) 145 (b) 289 (c) 217 (d) 433
  16. 16.What are the roots of x² − 11x + 30 = 0? (a) 5 and 6 (b) 2 and 15 (c) 1 and 10 (d) 1 and 30
  17. 17.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)
  18. 18.If a + b = 7 and ab = 12, what is a² + b²? (a) 73 (b) 37 (c) 49 (d) 25
  19. 19.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 63) (c) (x + 1)(x + 15) (d) (x + 7)(x + 9)
  20. 20.What is the product of the roots of x² − 9x + 20 = 0? (a) 20 (b) 9 (c) 21 (d) 19
  21. 21.If a + b = 19 and ab = 90, what is a² + b²? (a) 271 (b) 361 (c) 541 (d) 181
  22. 22.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 11035 (b) 9025 (c) 11025 (d) 10025
  23. 23.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 3625 (b) 4235 (c) 3025 (d) 4225
  24. 24.What is the sum of the roots of x² − 11x + 30 = 0? (a) 12 (b) 10 (c) 11 (d) 30
  25. 25.What is the sum of the roots of x² − 12x + 27 = 0? (a) 13 (b) 11 (c) 12 (d) 27
  26. 26.What is the sum of the roots of x² − 7x + 12 = 0? (a) 6 (b) 7 (c) 8 (d) 12
  27. 27.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 635 (b) 1225 (c) 625 (d) 875
  28. 28.If a + b = 20 and ab = 99, what is (a − b)²? (a) 2 (b) 202 (c) 4 (d) 400
  29. 29.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x − 5)(x − 6) (c) (x + 1)(x + 30) (d) (x + 5)(x + 6)
  30. 30.If a + b = 17 and ab = 66, what is (a − b)²? (a) 157 (b) 5 (c) 289 (d) 25
  31. 31.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)
  32. 32.If a + b = 14 and ab = 40, what is a² + b²? (a) 156 (b) 116 (c) 276 (d) 196
  33. 33.What is the sum of the roots of x² − 9x + 20 = 0? (a) 8 (b) 9 (c) 20 (d) 10
  34. 34.If a + b = 5 and ab = 4, what is (a − b)²? (a) 3 (b) 17 (c) 25 (d) 9
  35. 35.What is the sum of the roots of x² − 16x + 63 = 0? (a) 63 (b) 17 (c) 16 (d) 15
  36. 36.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 256 (b) 576 (c) 586 (d) 416
  37. 37.Which is the correct factorisation of x² + 11x + 24? (a) (x + 3)(x + 8) (b) (x + 1)(x + 10) (c) (x + 1)(x + 24) (d) (x − 3)(x − 8)
  38. 38.If a + b = 6 and ab = 8, what is (a − b)²? (a) 20 (b) 4 (c) 36 (d) 2
  39. 39.If a + b = 4 and ab = 3, what is a² + b²? (a) 16 (b) 13 (c) 10 (d) 22
  40. 40.If a + b = 13 and ab = 30, what is a² + b²? (a) 109 (b) 229 (c) 139 (d) 169

Answer key

  1. (x + 7)(x + 8)
  2. (x + 8)(x + 9)
  3. 160
  4. 2304
  5. (x + 2)(x + 4)
  6. 1369
  7. 202
  8. (x + 3)(x + 9)
  9. 85
  10. 74
  11. 16
  12. 441
  13. 3 and 5
  14. 17
  15. 145
  16. 5 and 6
  17. (x + 4)(x + 8)
  18. 25
  19. (x + 7)(x + 9)
  20. 20
  21. 181
  22. 11025
  23. 4225
  24. 11
  25. 12
  26. 7
  27. 625
  28. 4
  29. (x + 5)(x + 6)
  30. 25
  31. (x + 5)(x + 9)
  32. 116
  33. 9
  34. 9
  35. 16
  36. 576
  37. (x + 3)(x + 8)
  38. 4
  39. 10
  40. 109

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