Cubic Equations

Solution To The Depressed Cubic By William Greco

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A depressed cubic is an algebraic expression that is a third degree equation which lacks it’s second degree quadratic term. The equation shown below is an example of a depressed cubic. the depressed  ax 3  bx  c cubic

The main goal of this report is to solve for the x variable of the depressed cubic and arrive at a general solution for values of x, excluding complex numbers. One solution of x follows: subtract ax 3  bx  c  ax 3  bx  c  c  c  ax 3  bx  c  0 c  b  substitute  y    for x   3ay  3   b     b  ay   y      c  0    3ay 3ay       b3 simplify ay3  c  0 27a 2 y3 b3 0 27au multiply  27au 2  27acu  b3  0 by 27au

substitute u  ay3

u-c

3 2 1 27 a 4b  27ac  27ac  u   54 a 54a apply quadratic equation

3 2 1 27 a 4b  27ac   c   u    54 a  2 divide 27ac 54a

3 2 1 27 a 4b  27ac  c  u   54 a 2   a  a

multiply

 u 3 2



27 a 4b  27ac 1 by 54 a



27 a 4b3  27ac2  c  54a 2

33 a 4b3  27ac 2  c  u   54a 2 factor 27a

3 3a 4b3  27ac 2  c  u   54a 2 a 2b  a b

Solution To The Depressed Cubic By William Greco

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3a 4b3  27ac 2  c  u   18a 2 divide 18 3 54

12ab3  81a 2 c 2 c  18a 2

multiply  u 3a 4b3  27ac2





c 81a 2 c 2  12ab3  u   2 18a c 81a 2 c 2  12ab3 definition of u 3  ay   ay3 2 18a re-order terms

c 81a 2 c 2  12ab3  a y3 divide both 18a   2 sides by a a a 9a 9ac  81a 2 c 2  12ab3 common 3 2 18a 9a c   9ac  y  denominator 18a a 9ac  81a 2 c 2  12ab3  1  9ac  81a 2 c 2  12ab3 3 18a  y   y  18a 18a 2 a a multiply right hand side by 1 a

cube root  of both sides

y

3

3

y3 

3 3

b 3

y

3

3

9ac  81a 2 c 2  12ab3 18a 2

9ac  81a 2 c 2  12ab3 18a 2

  y a 3a 3

2

9ac  81a 2 c 2  12ab3 3

b

18a 2

9ac  81a 2 c 2  12ab3 3

18a

2

x y -    3a y 

3 subtract  x  b  -  from both sides  3a 

original definition of y   b 

9ac  81a 2 c 2  12ab3 3

18a 2



b  3 9ac  81a 2 c 2  12ab3 3a  3  18a 2 

   

Solution To The Depressed Cubic By William Greco

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x 3

x

9ac  81a 2 c 2  12ab3 3

18a 2



b  3 9ac  81a 2 c 2  12ab3 3a  3  18a 2 

equation-1a 

   

and 3

x

9ac  81a 2 c2  12ab3 3

18a

2



b  3 9ac  81a 2 c 2  12ab3 3a  3  18a 2 

equation-1b 

   

check  a  3, b  7, c  14 by equation-1a 3

x

2

2

9 314   813 14   12 373 3

2



18 3

7 3 2 2 3  9 314  813  14   12 37 3 3 2 3 18 3  

    

 1.221

by equation-1b 3

x

2

2

9 314   813 14   12 373 3

18 3

ax3  bx  c 3

2

3 1.221  7 1.221  14



7 3 2 2 3  9 314  813 14   12 37 3 3 2 3 18 3  

    

 1.221

Solution To The Depressed Cubic By William Greco

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by equation-1a 3

2

2

9 314   813 14   12 373 3

2

 1.683

18 3

7 3 2 2 3  9 314  813 14   12 37 3 3 2 3 18 3  

    

 0.462

1.683  0.462  1.221 by equation-1b 3

x

2

2

9 3 14   813  14   12 373 3

7

 0.462

2

3 2 2 3  9 314  813 14   12 37 3 3 2 3 18 3  

18 3

0.462  1.683  1.221 Other forms of the equation include but are not limited to the following: 1

 27ac 2  4b3 3   c  a  x    2a  6 3a    

b 1

 27ac 2  4b 3 3   c a 3a     2a  6 3a     1  

  3  1c 1  3  1 b 3 2  x=      4b  27c a  3   1   2  a  18  3     2  3  a     1  c  1  3  a       4b3  27c 2 a  3           2  a  18  a 2        1   3 2 2   4b  27c a    1    a2 x= 108c  12 3  6a  a     

William Greco 2404 Greensward N. Warrington, Pa. 18976

1 3

2

[email protected]

b 1   3 2 2   4b  27c a       a2  108c  12 3  a      

1 3

    

  1.683