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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 463 · 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 24²? (a) 256 (b) 586 (c) 576 (d) 416
  2. 2.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 8) (b) (x − 4)(x − 5) (c) (x + 4)(x + 5) (d) (x + 1)(x + 20)
  3. 3.If a + b = 14 and ab = 48, what is a² + b²? (a) 196 (b) 292 (c) 148 (d) 100
  4. 4.What are the roots of x² − 10x + 24 = 0? (a) 4 and 6 (b) 2 and 12 (c) 1 and 9 (d) 1 and 24
  5. 5.What are the roots of x² − 7x + 10 = 0? (a) 1 and 10 (b) 2 and 5 (c) 3 and 4 (d) 1 and 6
  6. 6.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 16 (c) 14 (d) 15
  7. 7.What are the roots of x² − 14x + 45 = 0? (a) 1 and 13 (b) 1 and 45 (c) 5 and 9 (d) 3 and 15
  8. 8.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1369 (b) 1859 (c) 1609 (d) 1849
  9. 9.If a + b = 19 and ab = 90, what is a² + b²? (a) 271 (b) 541 (c) 181 (d) 361
  10. 10.If a + b = 8 and ab = 15, what is a² + b²? (a) 94 (b) 34 (c) 64 (d) 49
  11. 11.If a + b = 16 and ab = 63, what is a² + b²? (a) 382 (b) 193 (c) 130 (d) 256
  12. 12.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2116 (b) 2916 (c) 2126 (d) 2484
  13. 13.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 529 (b) 391 (c) 289 (d) 299
  14. 14.If a + b = 16 and ab = 55, what is a² + b²? (a) 201 (b) 146 (c) 366 (d) 256
  15. 15.If a + b = 17 and ab = 72, what is (a − b)²? (a) 217 (b) 289 (c) 1 (d) 145
  16. 16.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 10016 (b) 9216 (c) 10826 (d) 10816
  17. 17.What is the sum of the roots of x² − 13x + 36 = 0? (a) 36 (b) 14 (c) 13 (d) 12
  18. 18.If a + b = 16 and ab = 63, what is (a − b)²? (a) 130 (b) 4 (c) 256 (d) 2
  19. 19.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x + 6)(x + 7) (c) (x − 6)(x − 7) (d) (x + 1)(x + 12)
  20. 20.What are the roots of x² − 16x + 63 = 0? (a) 1 and 63 (b) 1 and 15 (c) 3 and 21 (d) 7 and 9
  21. 21.Which is the correct factorisation of x² + 10x + 24? (a) (x − 4)(x − 6) (b) (x + 1)(x + 9) (c) (x + 4)(x + 6) (d) (x + 1)(x + 24)
  22. 22.Which is the correct factorisation of x² + 11x + 24? (a) (x + 3)(x + 8) (b) (x + 1)(x + 10) (c) (x − 3)(x − 8) (d) (x + 1)(x + 24)
  23. 23.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 1225 (b) 2025 (c) 1625 (d) 2035
  24. 24.What is the sum of the roots of x² − 11x + 28 = 0? (a) 10 (b) 12 (c) 28 (d) 11
  25. 25.What is the sum of the roots of x² − 6x + 8 = 0? (a) 7 (b) 6 (c) 5 (d) 8
  26. 26.What are the roots of x² − 13x + 40 = 0? (a) 1 and 40 (b) 1 and 12 (c) 2 and 20 (d) 5 and 8
  27. 27.What are the roots of x² − 9x + 14 = 0? (a) 1 and 14 (b) 2 and 7 (c) 1 and 8 (d) 3 and 6
  28. 28.What is the sum of the roots of x² − 10x + 16 = 0? (a) 9 (b) 10 (c) 16 (d) 11
  29. 29.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x − 6)(x − 8) (c) (x + 6)(x + 8) (d) (x + 1)(x + 13)
  30. 30.What is the product of the roots of x² − 12x + 35 = 0? (a) 36 (b) 35 (c) 34 (d) 12
  31. 31.If a + b = 9 and ab = 20, what is a² + b²? (a) 41 (b) 61 (c) 121 (d) 81
  32. 32.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 7235 (b) 5625 (c) 7225 (d) 6425
  33. 33.What are the roots of x² − 17x + 72 = 0? (a) 8 and 9 (b) 1 and 72 (c) 2 and 36 (d) 1 and 16
  34. 34.What is the sum of the roots of x² − 9x + 20 = 0? (a) 9 (b) 20 (c) 10 (d) 8
  35. 35.Which is the correct factorisation of x² + 7x + 12? (a) (x + 3)(x + 4) (b) (x + 1)(x + 12) (c) (x + 1)(x + 6) (d) (x − 3)(x − 4)
  36. 36.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9996 (b) 9614 (c) 10404 (d) 9604
  37. 37.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
  38. 38.Which is the correct factorisation of x² + 5x + 6? (a) (x + 2)(x + 3) (b) (x + 1)(x + 4) (c) (x + 1)(x + 6) (d) (x − 2)(x − 3)
  39. 39.What are the roots of x² − 11x + 30 = 0? (a) 5 and 6 (b) 1 and 10 (c) 1 and 30 (d) 2 and 15
  40. 40.What is the sum of the roots of x² − 11x + 18 = 0? (a) 10 (b) 18 (c) 11 (d) 12

Answer key

  1. 576
  2. (x + 4)(x + 5)
  3. 100
  4. 4 and 6
  5. 2 and 5
  6. 15
  7. 5 and 9
  8. 1849
  9. 181
  10. 34
  11. 130
  12. 2116
  13. 289
  14. 146
  15. 1
  16. 10816
  17. 13
  18. 4
  19. (x + 6)(x + 7)
  20. 7 and 9
  21. (x + 4)(x + 6)
  22. (x + 3)(x + 8)
  23. 2025
  24. 11
  25. 6
  26. 5 and 8
  27. 2 and 7
  28. 10
  29. (x + 6)(x + 8)
  30. 35
  31. 41
  32. 7225
  33. 8 and 9
  34. 9
  35. (x + 3)(x + 4)
  36. 9604
  37. 3 and 5
  38. (x + 2)(x + 3)
  39. 5 and 6
  40. 11

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