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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 986 · 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.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3374 (b) 3364 (c) 3844 (d) 3596
  2. 2.What is the sum of the roots of x² − 10x + 24 = 0? (a) 24 (b) 9 (c) 10 (d) 11
  3. 3.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 9) (b) (x − 2)(x − 8) (c) (x + 2)(x + 8) (d) (x + 1)(x + 16)
  4. 4.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6561 (b) 6241 (c) 6401 (d) 6571
  5. 5.Which is the correct factorisation of x² + 6x + 8? (a) (x + 2)(x + 4) (b) (x + 1)(x + 8) (c) (x − 2)(x − 4) (d) (x + 1)(x + 5)
  6. 6.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 15) (b) (x + 1)(x + 7) (c) (x − 3)(x − 5) (d) (x + 3)(x + 5)
  7. 7.If a + b = 13 and ab = 42, what is a² + b²? (a) 169 (b) 85 (c) 127 (d) 253
  8. 8.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x − 6)(x − 8) (c) (x + 6)(x + 8) (d) (x + 1)(x + 48)
  9. 9.What is the sum of the roots of x² − 11x + 28 = 0? (a) 11 (b) 28 (c) 10 (d) 12
  10. 10.What is the sum of the roots of x² − 9x + 14 = 0? (a) 9 (b) 10 (c) 14 (d) 8
  11. 11.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 9811 (b) 9801 (c) 9999 (d) 10201
  12. 12.If a + b = 13 and ab = 12, what is (a − b)²? (a) 11 (b) 169 (c) 145 (d) 121
  13. 13.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 529 (b) 289 (c) 299 (d) 391
  14. 14.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 40) (b) (x − 5)(x − 8) (c) (x + 5)(x + 8) (d) (x + 1)(x + 12)
  15. 15.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 1)(x + 63) (c) (x + 7)(x + 9) (d) (x − 7)(x − 9)
  16. 16.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 5329 (b) 4891 (c) 4489 (d) 4499
  17. 17.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 9419 (b) 9409 (c) 10609 (d) 9991
  18. 18.If a + b = 18 and ab = 80, what is a² + b²? (a) 484 (b) 324 (c) 164 (d) 244
  19. 19.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 4624 (b) 4896 (c) 5184 (d) 4634
  20. 20.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 6) (b) (x + 2)(x + 5) (c) (x + 1)(x + 10) (d) (x − 2)(x − 5)
  21. 21.Which is the correct factorisation of x² + 5x + 6? (a) (x + 2)(x + 3) (b) (x − 2)(x − 3) (c) (x + 1)(x + 4) (d) (x + 1)(x + 6)
  22. 22.What is the sum of the roots of x² − 12x + 27 = 0? (a) 12 (b) 27 (c) 11 (d) 13
  23. 23.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1859 (b) 1609 (c) 1369 (d) 1849
  24. 24.What is the product of the roots of x² − 6x + 8 = 0? (a) 9 (b) 8 (c) 7 (d) 6
  25. 25.If a + b = 17 and ab = 66, what is a² + b²? (a) 223 (b) 421 (c) 157 (d) 289
  26. 26.If a + b = 20 and ab = 96, what is (a − b)²? (a) 400 (b) 208 (c) 16 (d) 4
  27. 27.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2209 (b) 2809 (c) 2219 (d) 2491
  28. 28.If a + b = 18 and ab = 77, what is (a − b)²? (a) 324 (b) 170 (c) 16 (d) 4
  29. 29.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 401 (b) 361 (c) 451 (d) 441
  30. 30.If a + b = 19 and ab = 88, what is a² + b²? (a) 185 (b) 273 (c) 537 (d) 361
  31. 31.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 12) (b) (x − 4)(x − 9) (c) (x + 1)(x + 36) (d) (x + 4)(x + 9)
  32. 32.If a + b = 5 and ab = 4, what is (a − b)²? (a) 9 (b) 17 (c) 3 (d) 25
  33. 33.What is the product of the roots of x² − 12x + 32 = 0? (a) 12 (b) 33 (c) 31 (d) 32
  34. 34.If a + b = 16 and ab = 63, what is a² + b²? (a) 193 (b) 382 (c) 256 (d) 130
  35. 35.If a + b = 19 and ab = 84, what is (a − b)²? (a) 25 (b) 193 (c) 5 (d) 361
  36. 36.If a + b = 18 and ab = 77, what is a² + b²? (a) 247 (b) 324 (c) 478 (d) 170
  37. 37.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8291 (b) 8281 (c) 8101 (d) 7921
  38. 38.What are the roots of x² − 12x + 27 = 0? (a) 1 and 27 (b) 1 and 11 (c) 2 and 10 (d) 3 and 9
  39. 39.What is the sum of the roots of x² − 16x + 63 = 0? (a) 15 (b) 63 (c) 17 (d) 16
  40. 40.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 529 (b) 539 (c) 289 (d) 409

Answer key

  1. 3364
  2. 10
  3. (x + 2)(x + 8)
  4. 6561
  5. (x + 2)(x + 4)
  6. (x + 3)(x + 5)
  7. 85
  8. (x + 6)(x + 8)
  9. 11
  10. 9
  11. 9801
  12. 121
  13. 289
  14. (x + 5)(x + 8)
  15. (x + 7)(x + 9)
  16. 4489
  17. 9409
  18. 164
  19. 4624
  20. (x + 2)(x + 5)
  21. (x + 2)(x + 3)
  22. 12
  23. 1849
  24. 8
  25. 157
  26. 16
  27. 2209
  28. 16
  29. 441
  30. 185
  31. (x + 4)(x + 9)
  32. 9
  33. 32
  34. 130
  35. 25
  36. 170
  37. 8281
  38. 3 and 9
  39. 16
  40. 529

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