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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 802 · 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 25²? (a) 1225 (b) 875 (c) 635 (d) 625
  2. 2.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)
  3. 3.What is the sum of the roots of x² − 12x + 27 = 0? (a) 27 (b) 12 (c) 11 (d) 13
  4. 4.What is the sum of the roots of x² − 16x + 63 = 0? (a) 17 (b) 15 (c) 16 (d) 63
  5. 5.What are the roots of x² − 7x + 10 = 0? (a) 2 and 5 (b) 1 and 6 (c) 1 and 10 (d) 3 and 4
  6. 6.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 8)(x + 9) (c) (x − 8)(x − 9) (d) (x + 1)(x + 72)
  7. 7.What are the roots of x² − 9x + 20 = 0? (a) 1 and 20 (b) 1 and 8 (c) 2 and 10 (d) 4 and 5
  8. 8.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2916 (b) 2126 (c) 2484 (d) 2116
  9. 9.If a + b = 9 and ab = 14, what is (a − b)²? (a) 5 (b) 25 (c) 81 (d) 53
  10. 10.What is the product of the roots of x² − 16x + 63 = 0? (a) 64 (b) 62 (c) 63 (d) 16
  11. 11.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 9) (b) (x − 2)(x − 8) (c) (x + 1)(x + 16) (d) (x + 2)(x + 8)
  12. 12.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 3616 (b) 4096 (c) 3136 (d) 4106
  13. 13.If a + b = 7 and ab = 6, what is a² + b²? (a) 37 (b) 43 (c) 61 (d) 49
  14. 14.If a + b = 19 and ab = 84, what is (a − b)²? (a) 361 (b) 193 (c) 5 (d) 25
  15. 15.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 7)(x + 8) (c) (x − 7)(x − 8) (d) (x + 1)(x + 14)
  16. 16.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 9) (b) (x + 4)(x + 6) (c) (x − 4)(x − 6) (d) (x + 1)(x + 24)
  17. 17.If a + b = 12 and ab = 20, what is a² + b²? (a) 104 (b) 124 (c) 144 (d) 184
  18. 18.What is the sum of the roots of x² − 14x + 45 = 0? (a) 15 (b) 45 (c) 14 (d) 13
  19. 19.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2496 (b) 2704 (c) 2304 (d) 2314
  20. 20.What are the roots of x² − 13x + 42 = 0? (a) 1 and 42 (b) 6 and 7 (c) 2 and 21 (d) 1 and 12
  21. 21.What are the roots of x² − 9x + 18 = 0? (a) 2 and 9 (b) 3 and 6 (c) 1 and 8 (d) 1 and 18
  22. 22.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10009 (b) 10619 (c) 10609 (d) 9409
  23. 23.What are the roots of x² − 15x + 54 = 0? (a) 1 and 14 (b) 2 and 27 (c) 1 and 54 (d) 6 and 9
  24. 24.If a + b = 5 and ab = 6, what is (a − b)²? (a) 1 (b) 19 (c) 13 (d) 25
  25. 25.If a + b = 17 and ab = 72, what is a² + b²? (a) 433 (b) 289 (c) 217 (d) 145
  26. 26.What are the roots of x² − 12x + 35 = 0? (a) 1 and 11 (b) 2 and 10 (c) 1 and 35 (d) 5 and 7
  27. 27.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 334 (b) 396 (c) 324 (d) 484
  28. 28.If a + b = 7 and ab = 12, what is a² + b²? (a) 25 (b) 73 (c) 49 (d) 37
  29. 29.If a + b = 16 and ab = 63, what is a² + b²? (a) 130 (b) 382 (c) 193 (d) 256
  30. 30.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)
  31. 31.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)
  32. 32.Which is the correct factorisation of x² + 7x + 10? (a) (x − 2)(x − 5) (b) (x + 2)(x + 5) (c) (x + 1)(x + 6) (d) (x + 1)(x + 10)
  33. 33.If a + b = 21 and ab = 108, what is (a − b)²? (a) 3 (b) 441 (c) 225 (d) 9
  34. 34.What is the product of the roots of x² − 10x + 24 = 0? (a) 23 (b) 25 (c) 24 (d) 10
  35. 35.What are the roots of x² − 14x + 48 = 0? (a) 1 and 48 (b) 6 and 8 (c) 2 and 24 (d) 1 and 13
  36. 36.If a + b = 10 and ab = 24, what is a² + b²? (a) 100 (b) 52 (c) 148 (d) 76
  37. 37.If a + b = 3 and ab = 2, what is (a − b)²? (a) 5 (b) 7 (c) 1 (d) 9
  38. 38.Which is the correct factorisation of x² + 6x + 8? (a) (x − 2)(x − 4) (b) (x + 1)(x + 5) (c) (x + 2)(x + 4) (d) (x + 1)(x + 8)
  39. 39.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 451 (b) 361 (c) 441 (d) 401
  40. 40.If a + b = 13 and ab = 36, what is (a − b)²? (a) 5 (b) 97 (c) 169 (d) 25

Answer key

  1. 625
  2. (x + 3)(x + 5)
  3. 12
  4. 16
  5. 2 and 5
  6. (x + 8)(x + 9)
  7. 4 and 5
  8. 2116
  9. 25
  10. 63
  11. (x + 2)(x + 8)
  12. 4096
  13. 37
  14. 25
  15. (x + 7)(x + 8)
  16. (x + 4)(x + 6)
  17. 104
  18. 14
  19. 2304
  20. 6 and 7
  21. 3 and 6
  22. 10609
  23. 6 and 9
  24. 1
  25. 145
  26. 5 and 7
  27. 324
  28. 25
  29. 130
  30. (x + 6)(x + 8)
  31. (x + 6)(x + 7)
  32. (x + 2)(x + 5)
  33. 9
  34. 24
  35. 6 and 8
  36. 52
  37. 1
  38. (x + 2)(x + 4)
  39. 441
  40. 25

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