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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 456 · 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 − 13x + 42 = 0? (a) 14 (b) 13 (c) 12 (d) 42
  2. 2.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 6)(x + 7) (c) (x + 1)(x + 42) (d) (x + 1)(x + 12)
  3. 3.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 16 (c) 17 (d) 18
  4. 4.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 371 (b) 361 (c) 399 (d) 441
  5. 5.If a + b = 3 and ab = 2, what is (a − b)²? (a) 7 (b) 1 (c) 5 (d) 9
  6. 6.If a + b = 5 and ab = 6, what is a² + b²? (a) 25 (b) 19 (c) 37 (d) 13
  7. 7.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8109 (b) 8649 (c) 8659 (d) 7569
  8. 8.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1591 (b) 1369 (c) 1849 (d) 1379
  9. 9.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1444 (b) 1604 (c) 1774 (d) 1764
  10. 10.Which is the correct factorisation of x² + 10x + 16? (a) (x + 2)(x + 8) (b) (x + 1)(x + 9) (c) (x − 2)(x − 8) (d) (x + 1)(x + 16)
  11. 11.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 7) (b) (x + 1)(x + 12) (c) (x − 2)(x − 6) (d) (x + 2)(x + 6)
  12. 12.If a + b = 20 and ab = 99, what is (a − b)²? (a) 400 (b) 4 (c) 2 (d) 202
  13. 13.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 8)(x + 9) (c) (x − 8)(x − 9) (d) (x + 1)(x + 16)
  14. 14.What is the sum of the roots of x² − 16x + 63 = 0? (a) 15 (b) 17 (c) 16 (d) 63
  15. 15.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 15) (c) (x + 7)(x + 9) (d) (x + 1)(x + 63)
  16. 16.If a + b = 14 and ab = 40, what is a² + b²? (a) 116 (b) 276 (c) 196 (d) 156
  17. 17.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x − 5)(x − 7) (c) (x + 1)(x + 35) (d) (x + 5)(x + 7)
  18. 18.What are the roots of x² − 9x + 14 = 0? (a) 1 and 8 (b) 3 and 6 (c) 1 and 14 (d) 2 and 7
  19. 19.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 56) (c) (x + 1)(x + 14) (d) (x + 7)(x + 8)
  20. 20.If a + b = 19 and ab = 90, what is (a − b)²? (a) 271 (b) 181 (c) 361 (d) 1
  21. 21.If a + b = 17 and ab = 72, what is a² + b²? (a) 145 (b) 433 (c) 289 (d) 217
  22. 22.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 7) (b) (x + 3)(x + 5) (c) (x + 1)(x + 15) (d) (x − 3)(x − 5)
  23. 23.If a + b = 19 and ab = 90, what is a² + b²? (a) 181 (b) 541 (c) 271 (d) 361
  24. 24.Which is the correct factorisation of x² + 11x + 30? (a) (x + 5)(x + 6) (b) (x + 1)(x + 30) (c) (x + 1)(x + 10) (d) (x − 5)(x − 6)
  25. 25.If a + b = 7 and ab = 10, what is a² + b²? (a) 49 (b) 69 (c) 39 (d) 29
  26. 26.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 45) (b) (x + 5)(x + 9) (c) (x − 5)(x − 9) (d) (x + 1)(x + 13)
  27. 27.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6404 (b) 6734 (c) 6724 (d) 6084
  28. 28.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 961 (b) 899 (c) 851 (d) 841
  29. 29.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 11) (b) (x + 1)(x + 32) (c) (x + 4)(x + 8) (d) (x − 4)(x − 8)
  30. 30.What is the sum of the roots of x² − 14x + 48 = 0? (a) 13 (b) 14 (c) 48 (d) 15
  31. 31.What are the roots of x² − 11x + 28 = 0? (a) 4 and 7 (b) 1 and 28 (c) 1 and 10 (d) 2 and 14
  32. 32.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 18) (b) (x − 3)(x − 6) (c) (x + 1)(x + 8) (d) (x + 3)(x + 6)
  33. 33.If a + b = 14 and ab = 33, what is (a − b)²? (a) 196 (b) 8 (c) 130 (d) 64
  34. 34.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2611 (b) 2501 (c) 2401 (d) 2601
  35. 35.Which is the correct factorisation of x² + 13x + 40? (a) (x − 5)(x − 8) (b) (x + 1)(x + 40) (c) (x + 5)(x + 8) (d) (x + 1)(x + 12)
  36. 36.What are the roots of x² − 14x + 48 = 0? (a) 1 and 48 (b) 1 and 13 (c) 6 and 8 (d) 2 and 24
  37. 37.What is the sum of the roots of x² − 8x + 15 = 0? (a) 8 (b) 7 (c) 15 (d) 9
  38. 38.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 21) (b) (x + 3)(x + 7) (c) (x + 1)(x + 9) (d) (x − 3)(x − 7)
  39. 39.What is the sum of the roots of x² − 12x + 32 = 0? (a) 12 (b) 32 (c) 13 (d) 11
  40. 40.What are the roots of x² − 17x + 72 = 0? (a) 2 and 36 (b) 1 and 16 (c) 8 and 9 (d) 1 and 72

Answer key

  1. 13
  2. (x + 6)(x + 7)
  3. 17
  4. 361
  5. 1
  6. 13
  7. 8649
  8. 1369
  9. 1764
  10. (x + 2)(x + 8)
  11. (x + 2)(x + 6)
  12. 4
  13. (x + 8)(x + 9)
  14. 16
  15. (x + 7)(x + 9)
  16. 116
  17. (x + 5)(x + 7)
  18. 2 and 7
  19. (x + 7)(x + 8)
  20. 1
  21. 145
  22. (x + 3)(x + 5)
  23. 181
  24. (x + 5)(x + 6)
  25. 29
  26. (x + 5)(x + 9)
  27. 6724
  28. 841
  29. (x + 4)(x + 8)
  30. 14
  31. 4 and 7
  32. (x + 3)(x + 6)
  33. 64
  34. 2601
  35. (x + 5)(x + 8)
  36. 6 and 8
  37. 8
  38. (x + 3)(x + 7)
  39. 12
  40. 8 and 9

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