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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 874 · 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 + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 1)(x + 48) (c) (x + 6)(x + 8) (d) (x + 1)(x + 13)
  2. 2.Which is the correct factorisation of x² + 9x + 20? (a) (x − 4)(x − 5) (b) (x + 4)(x + 5) (c) (x + 1)(x + 8) (d) (x + 1)(x + 20)
  3. 3.What is the product of the roots of x² − 8x + 15 = 0? (a) 16 (b) 14 (c) 8 (d) 15
  4. 4.If a + b = 12 and ab = 27, what is (a − b)²? (a) 6 (b) 144 (c) 36 (d) 90
  5. 5.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9975 (b) 9025 (c) 11025 (d) 9035
  6. 6.If a + b = 17 and ab = 66, what is (a − b)²? (a) 289 (b) 157 (c) 5 (d) 25
  7. 7.If a + b = 9 and ab = 18, what is (a − b)²? (a) 45 (b) 9 (c) 3 (d) 81
  8. 8.If a + b = 5 and ab = 4, what is a² + b²? (a) 33 (b) 25 (c) 17 (d) 21
  9. 9.What are the roots of x² − 9x + 18 = 0? (a) 1 and 8 (b) 1 and 18 (c) 2 and 9 (d) 3 and 6
  10. 10.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 45) (b) (x + 1)(x + 13) (c) (x + 5)(x + 9) (d) (x − 5)(x − 9)
  11. 11.What is the product of the roots of x² − 10x + 21 = 0? (a) 10 (b) 21 (c) 20 (d) 22
  12. 12.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x − 7)(x − 8) (c) (x + 7)(x + 8) (d) (x + 1)(x + 56)
  13. 13.If a + b = 20 and ab = 96, what is (a − b)²? (a) 208 (b) 16 (c) 400 (d) 4
  14. 14.If a + b = 13 and ab = 12, what is a² + b²? (a) 169 (b) 145 (c) 157 (d) 193
  15. 15.What is the product of the roots of x² − 11x + 24 = 0? (a) 23 (b) 24 (c) 25 (d) 11
  16. 16.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 324 (b) 396 (c) 334 (d) 484
  17. 17.If a + b = 5 and ab = 6, what is (a − b)²? (a) 25 (b) 13 (c) 1 (d) 19
  18. 18.What are the roots of x² − 12x + 35 = 0? (a) 5 and 7 (b) 1 and 35 (c) 1 and 11 (d) 2 and 10
  19. 19.If a + b = 10 and ab = 24, what is (a − b)²? (a) 52 (b) 2 (c) 4 (d) 100
  20. 20.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8101 (b) 8281 (c) 7921 (d) 8291
  21. 21.If a + b = 19 and ab = 84, what is (a − b)²? (a) 193 (b) 361 (c) 5 (d) 25
  22. 22.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3979 (b) 3969 (c) 3609 (d) 3249
  23. 23.If a + b = 12 and ab = 32, what is a² + b²? (a) 208 (b) 80 (c) 112 (d) 144
  24. 24.What is the sum of the roots of x² − 6x + 8 = 0? (a) 5 (b) 8 (c) 6 (d) 7
  25. 25.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)
  26. 26.If a + b = 19 and ab = 90, what is a² + b²? (a) 361 (b) 271 (c) 541 (d) 181
  27. 27.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 576 (b) 256 (c) 586 (d) 416
  28. 28.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1616 (b) 1936 (c) 1946 (d) 1296
  29. 29.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x − 7)(x − 9) (c) (x + 1)(x + 63) (d) (x + 1)(x + 15)
  30. 30.Which is the correct factorisation of x² + 9x + 18? (a) (x − 3)(x − 6) (b) (x + 1)(x + 18) (c) (x + 1)(x + 8) (d) (x + 3)(x + 6)
  31. 31.What is the sum of the roots of x² − 12x + 35 = 0? (a) 13 (b) 35 (c) 11 (d) 12
  32. 32.If a + b = 14 and ab = 40, what is (a − b)²? (a) 36 (b) 6 (c) 196 (d) 116
  33. 33.What are the roots of x² − 14x + 48 = 0? (a) 1 and 13 (b) 1 and 48 (c) 2 and 24 (d) 6 and 8
  34. 34.What is the product of the roots of x² − 13x + 42 = 0? (a) 41 (b) 42 (c) 43 (d) 13
  35. 35.What is the sum of the roots of x² − 16x + 63 = 0? (a) 17 (b) 63 (c) 15 (d) 16
  36. 36.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10009 (b) 10619 (c) 10609 (d) 9409
  37. 37.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10404 (b) 10414 (c) 10004 (d) 9604
  38. 38.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 875 (b) 635 (c) 1225 (d) 625
  39. 39.If a + b = 17 and ab = 60, what is a² + b²? (a) 409 (b) 289 (c) 229 (d) 169
  40. 40.What is the product of the roots of x² − 17x + 72 = 0? (a) 72 (b) 73 (c) 71 (d) 17

Answer key

  1. (x + 6)(x + 8)
  2. (x + 4)(x + 5)
  3. 15
  4. 36
  5. 9025
  6. 25
  7. 9
  8. 17
  9. 3 and 6
  10. (x + 5)(x + 9)
  11. 21
  12. (x + 7)(x + 8)
  13. 16
  14. 145
  15. 24
  16. 324
  17. 1
  18. 5 and 7
  19. 4
  20. 8281
  21. 25
  22. 3969
  23. 80
  24. 6
  25. (x + 8)(x + 9)
  26. 181
  27. 576
  28. 1936
  29. (x + 7)(x + 9)
  30. (x + 3)(x + 6)
  31. 12
  32. 36
  33. 6 and 8
  34. 42
  35. 16
  36. 10609
  37. 10404
  38. 625
  39. 169
  40. 72

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