On Breaking-down any Triangle into
a pair of Mirror-Imaged Equilateral Triangles

by Frank C. Fung ( 1st published in September, 2021. )

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Section XXI - Solving the 3rd-Order Polynomial Equation

Summary for the Section :

• In this Section , we shall make our first attempt to solve the { 3rd-Order Polynomial Equation } :
  • 

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Reformatting the 3rd-Order Polynomial Equation :

• Let us bring in again the { 3rd-Order Polynomial Equation } from the last Section , Section XX :
  • i.e. this equation :
    • 

  Let us now reformat this equation for our further analysis below ,

  • by introducing the fixed values [ M ] and [ N ] , as follows :
    • 

    • 

  And the new format of the equation is then :

  • 

• And we take special note here that :
  • Once the complex values [ A ] , [ B ] and [ C ] are fixed ,
    • the complex values [ K1 ] , [ KO ] , [ M ] and [ N ] will also be fixed accordingly .

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Setting up [ M ] & [ N ] in the usual { complex-number format } :

• Let us now set up [ M ] and [ N ] in the usual { complex-number format } via :
  • introducing the fixed real-value [ R-sub-M ] and the fixed angle [ THETA-sub-M ] ,
    • so that we can now express [ M ] in this format below :
      • 

        with :

        • 

  • introducing the fixed real-value [ R-sub-N ] and the fixed angle [ THETA-sub-N ] ,
    • so that we can now express [ N ] in this format below :
      • 

        with :

        • 

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Bringing in the Algebriac Identity :

• Let us now bring in an Algebriac Identity involving the Complex Variabes [ T ] and [ U ] .
  • We then have this expression for the { cube of ( [ T ] + [ U ] ) } :
    • 

    Consequently :

    • 

    And on re-arranging terms , we have :

    • 

• This then is the Algebraic Identity we shall be using below .

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Reconciling the 2 equations :

• Let us now bring back these 2 equations from above for a quick comparison :
  • First , the reformatted { 3rd-Order Polynomial Equation } :
    • 

  • Secondly , the Algebraic Identity :
    • 

  And we take note here that :

  • IF :
    • these 2 particular conditions are fully met , i.e. :
      • Condition ONE :
        • 

      • Condition TWO :
        • 

    THEN :

    • for the particular set of { [ T ] and [ U ] } satisfying the 2 conditions ,
      • we can always set up a corresponding value of [ x ] given by :
        • 

        and consequently yielding :

        • 

    SO THAT :

    • the 2 equations above will match exactly numerical-wise :
      • for the particular set of values for { [ T ] , [ U ] and [ x ] } .

• We shall now proceed to find a set of values of [ T ] and [ U ] :
  • that satisfy both { Condition ONE } and { Condition TWO } ,

  so that :

  • a solution value of [ x ] can be obtained for the reformatted { 3rd-Order Polynomial Equation } via :
    • 

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Solving for [ T ] and [ U ] :

• Let us now bring back { Condition ONE } :
  • 

  On re-arranging terms , we have :

  • 

  And on cubing both sides of the equation , we have :

  • 

• Let us now bring back { Condition TWO } :
  • 

  Substituting therein the result we have immediately above then yields us this equation :

  • 

  And on re-arranging terms , we have :

  • 

• Let us now multiply throughout by [ T-cube ] to yield :
  • 

  And on adding [ N-square ] to both sides of the equation , we have :

  • 

  Consequently :

  • 

• Let us now take the square-root on both sides to yield :
  • 

  We then have :

  • 

    or alternately :

  • 

• We then set up the fixed-values [ W1 ] and [ W2 ] as follows :
  • 

  • 

  so that we can now re-write the previous 2 equations as :

  • 

    and

  • 

• And we take note here in passing that :
  • 

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Setting up [ W1 ] and [ W2 ] in the usual { complex number format } :

• Let us now set up the fixed real-value [ R-sub-W1 ] and the fixed angle [ THETA-sub-W1 ] ,
  • so that we can now express [ W1 ] in this format below :
    • 

      with :

      • 

• Let us also set up the fixed real-value [ R-sub-W2 ] and the fixed angle [ THETA-sub-W2 ] ,
  • so that we can now express [ W2 ] in this format below :
    • 

      with :

      • 

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What's Next :

• And with this in hand , we shall now proceed to :
  • solve for the solution values of [ x ] based on the 1st Equation above , i.e. this equation :
    • 

      in a later Section , namley Section XXII ;

    and

  • solve for the solution values of [ x ] based on the 2nd Equation above , i.e. this equation :
    • 

      in a still later Section , namley Section XXIII .

  But first , let us do a quick review of the { Multiplication of Complex Number } , next :

  • so that we can establish certain { complex-number relations } for our use in Sections to come .

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go to the next Section : Section XXI - A Quick Review of Multiplication of Complex Numbers

go to the last Section : Section XIX - Creating the 3rd-Order Polynomial Equation from { Triangle A-B-C }

return to the Pair of Mirror-Imaged Equilateral Triangles Homepage

Original dated 2021-9-07