X^7-X^6-X^5=0

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Solution for X^7-X^6-X^5=0 equation: 


Simplifying
X7 + -1X6 + -1X5 = 0

Reorder the terms:
-1X5 + -1X6 + X7 = 0

Solving
-1X5 + -1X6 + X7 = 0

Solving for variable 'X'.

Factor out the Greatest Common Factor (GCF), 'X5'.
X5(-1 + -1X + X2) = 0


Subproblem 1
Set the factor 'X5' equal to zero and attempt to solve:

Simplifying
X5 = 0

Solving
X5 = 0

Move all terms containing X to the left, all other terms to the right.

Simplifying
X5 = 0

The solution to this equation could not be determined.

This subproblem is being ignored because a solution could not be determined.

Subproblem 2
Set the factor '(-1 + -1X + X2)' equal to zero and attempt to solve:

Simplifying
-1 + -1X + X2 = 0

Solving
-1 + -1X + X2 = 0

Begin completing the square.

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -1X + 1 + X2 = 0 + 1

Reorder the terms:
-1 + 1 + -1X + X2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -1X + X2 = 0 + 1
-1X + X2 = 0 + 1

Combine like terms: 0 + 1 = 1
-1X + X2 = 1

The X term is -1X.  Take half its coefficient (-0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1X + 0.25 + X2 = 1 + 0.25

Reorder the terms:
0.25 + -1X + X2 = 1 + 0.25

Combine like terms: 1 + 0.25 = 1.25
0.25 + -1X + X2 = 1.25

Factor a perfect square on the left side:
(X + -0.5)(X + -0.5) = 1.25

Calculate the square root of the right side: 1.118033989

Break this problem into two subproblems by setting 
(X + -0.5) equal to 1.118033989 and -1.118033989.


Subproblem 1
X + -0.5 = 1.118033989

Simplifying
X + -0.5 = 1.118033989

Reorder the terms:
-0.5 + X = 1.118033989

Solving
-0.5 + X = 1.118033989

Solving for variable 'X'.

Move all terms containing X to the left, all other terms to the right.

Add '0.5' to each side of the equation.
-0.5 + 0.5 + X = 1.118033989 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + X = 1.118033989 + 0.5
X = 1.118033989 + 0.5

Combine like terms: 1.118033989 + 0.5 = 1.618033989
X = 1.618033989

Simplifying
X = 1.618033989


Subproblem 2
X + -0.5 = -1.118033989

Simplifying
X + -0.5 = -1.118033989

Reorder the terms:
-0.5 + X = -1.118033989

Solving
-0.5 + X = -1.118033989

Solving for variable 'X'.

Move all terms containing X to the left, all other terms to the right.

Add '0.5' to each side of the equation.
-0.5 + 0.5 + X = -1.118033989 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + X = -1.118033989 + 0.5
X = -1.118033989 + 0.5

Combine like terms: -1.118033989 + 0.5 = -0.618033989
X = -0.618033989

Simplifying
X = -0.618033989


Solution
The solution to the problem is based on the solutions
from the subproblems.
X = {1.618033989, -0.618033989}
Solution
X = {1.618033989, -0.618033989}

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