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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 40 · 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 product of the roots of x² − 7x + 12 = 0? (a) 13 (b) 7 (c) 12 (d) 11
  2. 2.What is the product of the roots of x² − 16x + 63 = 0? (a) 64 (b) 63 (c) 62 (d) 16
  3. 3.What is the product of the roots of x² − 8x + 12 = 0? (a) 11 (b) 12 (c) 8 (d) 13
  4. 4.Which is the correct factorisation of x² + 14x + 45? (a) (x − 5)(x − 9) (b) (x + 1)(x + 13) (c) (x + 1)(x + 45) (d) (x + 5)(x + 9)
  5. 5.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 4096 (b) 3616 (c) 4106 (d) 3136
  6. 6.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 24) (b) (x + 3)(x + 8) (c) (x + 1)(x + 10) (d) (x − 3)(x − 8)
  7. 7.What are the roots of x² − 12x + 35 = 0? (a) 1 and 11 (b) 1 and 35 (c) 2 and 10 (d) 5 and 7
  8. 8.If a + b = 20 and ab = 99, what is a² + b²? (a) 202 (b) 598 (c) 301 (d) 400
  9. 9.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 10816 (b) 9216 (c) 9226 (d) 9984
  10. 10.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 5776 (b) 6384 (c) 7056 (d) 5786
  11. 11.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 299 (b) 529 (c) 289 (d) 391
  12. 12.If a + b = 12 and ab = 32, what is (a − b)²? (a) 4 (b) 80 (c) 144 (d) 16
  13. 13.What is the sum of the roots of x² − 14x + 48 = 0? (a) 48 (b) 13 (c) 14 (d) 15
  14. 14.If a + b = 17 and ab = 72, what is (a − b)²? (a) 1 (b) 145 (c) 289 (d) 217
  15. 15.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 16) (c) (x + 8)(x + 9) (d) (x + 1)(x + 72)
  16. 16.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 10) (b) (x + 1)(x + 28) (c) (x + 4)(x + 7) (d) (x − 4)(x − 7)
  17. 17.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 6) (b) (x + 2)(x + 5) (c) (x − 2)(x − 5) (d) (x + 1)(x + 10)
  18. 18.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x + 1)(x + 14) (c) (x + 1)(x + 56) (d) (x − 7)(x − 8)
  19. 19.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1604 (b) 1764 (c) 1774 (d) 1444
  20. 20.Which is the correct factorisation of x² + 11x + 30? (a) (x + 5)(x + 6) (b) (x − 5)(x − 6) (c) (x + 1)(x + 10) (d) (x + 1)(x + 30)
  21. 21.Which is the correct factorisation of x² + 10x + 16? (a) (x − 2)(x − 8) (b) (x + 2)(x + 8) (c) (x + 1)(x + 16) (d) (x + 1)(x + 9)
  22. 22.If a + b = 9 and ab = 18, what is a² + b²? (a) 45 (b) 81 (c) 63 (d) 117
  23. 23.If a + b = 15 and ab = 56, what is (a − b)²? (a) 113 (b) 169 (c) 1 (d) 225
  24. 24.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 399 (b) 361 (c) 371 (d) 441
  25. 25.If a + b = 13 and ab = 42, what is (a − b)²? (a) 127 (b) 1 (c) 169 (d) 85
  26. 26.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 4499 (b) 4489 (c) 5329 (d) 4891
  27. 27.If a + b = 20 and ab = 96, what is (a − b)²? (a) 400 (b) 16 (c) 208 (d) 4
  28. 28.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 6889 (b) 5939 (c) 6391 (d) 5929
  29. 29.If a + b = 12 and ab = 20, what is a² + b²? (a) 124 (b) 104 (c) 144 (d) 184
  30. 30.If a + b = 17 and ab = 70, what is (a − b)²? (a) 3 (b) 289 (c) 9 (d) 149
  31. 31.What is the product of the roots of x² − 6x + 8 = 0? (a) 7 (b) 9 (c) 6 (d) 8
  32. 32.If a + b = 4 and ab = 3, what is (a − b)²? (a) 2 (b) 10 (c) 16 (d) 4
  33. 33.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 2035 (b) 3025 (c) 2475 (d) 2025
  34. 34.What are the roots of x² − 9x + 14 = 0? (a) 3 and 6 (b) 1 and 14 (c) 2 and 7 (d) 1 and 8
  35. 35.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x + 1)(x + 42) (c) (x − 6)(x − 7) (d) (x + 6)(x + 7)
  36. 36.What is the product of the roots of x² − 10x + 16 = 0? (a) 10 (b) 15 (c) 17 (d) 16
  37. 37.Which is the correct factorisation of x² + 13x + 36? (a) (x + 4)(x + 9) (b) (x + 1)(x + 36) (c) (x − 4)(x − 9) (d) (x + 1)(x + 12)
  38. 38.What is the product of the roots of x² − 14x + 48 = 0? (a) 49 (b) 14 (c) 48 (d) 47
  39. 39.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 18 (c) 17 (d) 16
  40. 40.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2916 (b) 2116 (c) 2926 (d) 2516

Answer key

  1. 12
  2. 63
  3. 12
  4. (x + 5)(x + 9)
  5. 4096
  6. (x + 3)(x + 8)
  7. 5 and 7
  8. 202
  9. 9216
  10. 5776
  11. 289
  12. 16
  13. 14
  14. 1
  15. (x + 8)(x + 9)
  16. (x + 4)(x + 7)
  17. (x + 2)(x + 5)
  18. (x + 7)(x + 8)
  19. 1764
  20. (x + 5)(x + 6)
  21. (x + 2)(x + 8)
  22. 45
  23. 1
  24. 361
  25. 1
  26. 4489
  27. 16
  28. 5929
  29. 104
  30. 9
  31. 8
  32. 4
  33. 2025
  34. 2 and 7
  35. (x + 6)(x + 7)
  36. 16
  37. (x + 4)(x + 9)
  38. 48
  39. 17
  40. 2916

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