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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 128 · 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 104²? (a) 9216 (b) 10016 (c) 10826 (d) 10816
  2. 2.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2116 (b) 2516 (c) 2926 (d) 2916
  3. 3.What is the sum of the roots of x² − 10x + 24 = 0? (a) 9 (b) 11 (c) 10 (d) 24
  4. 4.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x + 1)(x + 8) (c) (x − 4)(x − 5) (d) (x + 4)(x + 5)
  5. 5.What is the sum of the roots of x² − 11x + 18 = 0? (a) 10 (b) 11 (c) 12 (d) 18
  6. 6.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x − 5)(x − 6) (c) (x + 1)(x + 30) (d) (x + 5)(x + 6)
  7. 7.If a + b = 5 and ab = 4, what is (a − b)²? (a) 17 (b) 3 (c) 9 (d) 25
  8. 8.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)
  9. 9.If a + b = 13 and ab = 12, what is a² + b²? (a) 169 (b) 193 (c) 145 (d) 157
  10. 10.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)
  11. 11.What is the sum of the roots of x² − 8x + 15 = 0? (a) 9 (b) 7 (c) 15 (d) 8
  12. 12.What is the product of the roots of x² − 16x + 63 = 0? (a) 64 (b) 16 (c) 63 (d) 62
  13. 13.If a + b = 9 and ab = 20, what is a² + b²? (a) 121 (b) 81 (c) 61 (d) 41
  14. 14.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2714 (b) 2504 (c) 2704 (d) 2304
  15. 15.Which is the correct factorisation of x² + 12x + 35? (a) (x + 5)(x + 7) (b) (x − 5)(x − 7) (c) (x + 1)(x + 11) (d) (x + 1)(x + 35)
  16. 16.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)
  17. 17.If a + b = 19 and ab = 90, what is (a − b)²? (a) 361 (b) 271 (c) 181 (d) 1
  18. 18.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 5625 (b) 6375 (c) 7225 (d) 5635
  19. 19.What is the product of the roots of x² − 12x + 27 = 0? (a) 28 (b) 26 (c) 12 (d) 27
  20. 20.Which is the correct factorisation of x² + 13x + 42? (a) (x + 6)(x + 7) (b) (x + 1)(x + 42) (c) (x − 6)(x − 7) (d) (x + 1)(x + 12)
  21. 21.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 635 (b) 875 (c) 1225 (d) 625
  22. 22.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 7)(x + 8) (c) (x + 1)(x + 14) (d) (x + 1)(x + 56)
  23. 23.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 9801 (b) 10201 (c) 9811 (d) 9999
  24. 24.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 28) (b) (x + 4)(x + 7) (c) (x + 1)(x + 10) (d) (x − 4)(x − 7)
  25. 25.What is the product of the roots of x² − 8x + 15 = 0? (a) 8 (b) 14 (c) 16 (d) 15
  26. 26.If a + b = 21 and ab = 108, what is (a − b)²? (a) 441 (b) 3 (c) 225 (d) 9
  27. 27.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 9216 (b) 10816 (c) 9984 (d) 9226
  28. 28.Which is the correct factorisation of x² + 12x + 27? (a) (x − 3)(x − 9) (b) (x + 3)(x + 9) (c) (x + 1)(x + 27) (d) (x + 1)(x + 11)
  29. 29.What is the product of the roots of x² − 6x + 8 = 0? (a) 6 (b) 7 (c) 9 (d) 8
  30. 30.If a + b = 18 and ab = 80, what is a² + b²? (a) 484 (b) 244 (c) 324 (d) 164
  31. 31.If a + b = 20 and ab = 99, what is (a − b)²? (a) 4 (b) 202 (c) 2 (d) 400
  32. 32.If a + b = 8 and ab = 15, what is a² + b²? (a) 49 (b) 34 (c) 64 (d) 94
  33. 33.Which is the correct factorisation of x² + 10x + 24? (a) (x − 4)(x − 6) (b) (x + 1)(x + 24) (c) (x + 1)(x + 9) (d) (x + 4)(x + 6)
  34. 34.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1681 (b) 1521 (c) 1531 (d) 1599
  35. 35.What is the product of the roots of x² − 12x + 35 = 0? (a) 12 (b) 34 (c) 35 (d) 36
  36. 36.What is the sum of the roots of x² − 16x + 63 = 0? (a) 15 (b) 16 (c) 63 (d) 17
  37. 37.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3596 (b) 3364 (c) 3844 (d) 3374
  38. 38.If a + b = 15 and ab = 50, what is a² + b²? (a) 325 (b) 225 (c) 175 (d) 125
  39. 39.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2819 (b) 2509 (c) 2809 (d) 2209
  40. 40.If a + b = 5 and ab = 4, what is a² + b²? (a) 33 (b) 17 (c) 25 (d) 21

Answer key

  1. 10816
  2. 2916
  3. 10
  4. (x + 4)(x + 5)
  5. 11
  6. (x + 5)(x + 6)
  7. 9
  8. (x + 2)(x + 8)
  9. 145
  10. (x + 8)(x + 9)
  11. 8
  12. 63
  13. 41
  14. 2704
  15. (x + 5)(x + 7)
  16. (x + 7)(x + 9)
  17. 1
  18. 5625
  19. 27
  20. (x + 6)(x + 7)
  21. 625
  22. (x + 7)(x + 8)
  23. 9801
  24. (x + 4)(x + 7)
  25. 15
  26. 9
  27. 9216
  28. (x + 3)(x + 9)
  29. 8
  30. 164
  31. 4
  32. 34
  33. (x + 4)(x + 6)
  34. 1521
  35. 35
  36. 16
  37. 3364
  38. 125
  39. 2809
  40. 17

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