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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 905 · 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 − 11x + 28 = 0? (a) 27 (b) 11 (c) 28 (d) 29
  2. 2.If a + b = 19 and ab = 90, what is a² + b²? (a) 271 (b) 541 (c) 361 (d) 181
  3. 3.If a + b = 7 and ab = 10, what is (a − b)²? (a) 49 (b) 9 (c) 3 (d) 29
  4. 4.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 16 (c) 17 (d) 72
  5. 5.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 266 (b) 384 (c) 576 (d) 256
  6. 6.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10004 (b) 9604 (c) 10404 (d) 10414
  7. 7.What is the sum of the roots of x² − 11x + 24 = 0? (a) 10 (b) 12 (c) 11 (d) 24
  8. 8.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 8836 (b) 8116 (c) 7396 (d) 8846
  9. 9.If a + b = 20 and ab = 96, what is (a − b)²? (a) 208 (b) 4 (c) 16 (d) 400
  10. 10.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3721 (b) 3481 (c) 3601 (d) 3731
  11. 11.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x + 1)(x + 42) (c) (x + 6)(x + 7) (d) (x − 6)(x − 7)
  12. 12.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2491 (b) 2809 (c) 2219 (d) 2209
  13. 13.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 5635 (b) 5625 (c) 6375 (d) 7225
  14. 14.If a + b = 15 and ab = 56, what is (a − b)²? (a) 225 (b) 113 (c) 169 (d) 1
  15. 15.If a + b = 20 and ab = 96, what is a² + b²? (a) 400 (b) 208 (c) 304 (d) 592
  16. 16.If a + b = 11 and ab = 30, what is a² + b²? (a) 121 (b) 61 (c) 181 (d) 91
  17. 17.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 5184 (b) 4634 (c) 4896 (d) 4624
  18. 18.If a + b = 18 and ab = 80, what is a² + b²? (a) 484 (b) 164 (c) 244 (d) 324
  19. 19.If a + b = 5 and ab = 6, what is a² + b²? (a) 13 (b) 25 (c) 37 (d) 19
  20. 20.If a + b = 14 and ab = 45, what is (a − b)²? (a) 196 (b) 16 (c) 106 (d) 4
  21. 21.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 12) (b) (x − 5)(x − 8) (c) (x + 5)(x + 8) (d) (x + 1)(x + 40)
  22. 22.What is the sum of the roots of x² − 6x + 8 = 0? (a) 6 (b) 8 (c) 5 (d) 7
  23. 23.What is the product of the roots of x² − 15x + 54 = 0? (a) 53 (b) 55 (c) 54 (d) 15
  24. 24.If a + b = 17 and ab = 70, what is a² + b²? (a) 149 (b) 219 (c) 289 (d) 429
  25. 25.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x + 1)(x + 48) (c) (x + 6)(x + 8) (d) (x − 6)(x − 8)
  26. 26.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)
  27. 27.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 9801 (b) 9999 (c) 10201 (d) 9811
  28. 28.If a + b = 21 and ab = 108, what is (a − b)²? (a) 441 (b) 225 (c) 3 (d) 9
  29. 29.What is the sum of the roots of x² − 9x + 20 = 0? (a) 8 (b) 10 (c) 20 (d) 9
  30. 30.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9035 (b) 11025 (c) 9975 (d) 9025
  31. 31.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 529 (b) 539 (c) 409 (d) 289
  32. 32.Which is the correct factorisation of x² + 13x + 36? (a) (x + 4)(x + 9) (b) (x − 4)(x − 9) (c) (x + 1)(x + 12) (d) (x + 1)(x + 36)
  33. 33.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 35) (b) (x + 1)(x + 11) (c) (x − 5)(x − 7) (d) (x + 5)(x + 7)
  34. 34.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 12) (b) (x − 2)(x − 6) (c) (x + 1)(x + 7) (d) (x + 2)(x + 6)
  35. 35.Which is the correct factorisation of x² + 9x + 14? (a) (x − 2)(x − 7) (b) (x + 1)(x + 8) (c) (x + 2)(x + 7) (d) (x + 1)(x + 14)
  36. 36.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 484 (b) 396 (c) 324 (d) 334
  37. 37.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 4) (b) (x − 2)(x − 3) (c) (x + 1)(x + 6) (d) (x + 2)(x + 3)
  38. 38.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 1)(x + 72) (c) (x + 8)(x + 9) (d) (x − 8)(x − 9)
  39. 39.If a + b = 18 and ab = 72, what is a² + b²? (a) 252 (b) 180 (c) 468 (d) 324
  40. 40.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 14) (c) (x + 7)(x + 8) (d) (x + 1)(x + 56)

Answer key

  1. 28
  2. 181
  3. 9
  4. 17
  5. 256
  6. 10404
  7. 11
  8. 8836
  9. 16
  10. 3721
  11. (x + 6)(x + 7)
  12. 2209
  13. 5625
  14. 1
  15. 208
  16. 61
  17. 4624
  18. 164
  19. 13
  20. 16
  21. (x + 5)(x + 8)
  22. 6
  23. 54
  24. 149
  25. (x + 6)(x + 8)
  26. (x + 5)(x + 6)
  27. 9801
  28. 9
  29. 9
  30. 9025
  31. 529
  32. (x + 4)(x + 9)
  33. (x + 5)(x + 7)
  34. (x + 2)(x + 6)
  35. (x + 2)(x + 7)
  36. 324
  37. (x + 2)(x + 3)
  38. (x + 8)(x + 9)
  39. 180
  40. (x + 7)(x + 8)

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