z^6+2.5z^5+z^4=

Solution for z^6+2.5z^5+z^4= equation:

Simplifying
z6 + 2.5z5 + z4 = 0

Reorder the terms:
z4 + 2.5z5 + z6 = 0

Solving
z4 + 2.5z5 + z6 = 0

Solving for variable 'z'.

Factor out the Greatest Common Factor (GCF), 'z4'.
z4(1 + 2.5z + z2) = 0

Subproblem 1
Set the factor 'z4' equal to zero and attempt to solve:

Simplifying
z4 = 0

Solving
z4 = 0

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

Simplifying
z4 = 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 + 2.5z + z2)' equal to zero and attempt to solve:

Simplifying
1 + 2.5z + z2 = 0

Solving
1 + 2.5z + z2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + 2.5z + -1 + z2 = 0 + -1

Reorder the terms:
1 + -1 + 2.5z + z2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + 2.5z + z2 = 0 + -1
2.5z + z2 = 0 + -1

Combine like terms: 0 + -1 = -1
2.5z + z2 = -1

The z term is 2.5z.  Take half its coefficient (1.25).
Square it (1.5625) and add it to both sides.

Add '1.5625' to each side of the equation.
2.5z + 1.5625 + z2 = -1 + 1.5625

Reorder the terms:
1.5625 + 2.5z + z2 = -1 + 1.5625

Combine like terms: -1 + 1.5625 = 0.5625
1.5625 + 2.5z + z2 = 0.5625

Factor a perfect square on the left side:
(z + 1.25)(z + 1.25) = 0.5625

Calculate the square root of the right side: 0.75

Break this problem into two subproblems by setting 
(z + 1.25) equal to 0.75 and -0.75.

Subproblem 1
z + 1.25 = 0.75

Simplifying
z + 1.25 = 0.75

Reorder the terms:
1.25 + z = 0.75

Solving
1.25 + z = 0.75

Solving for variable 'z'.

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

Add '-1.25' to each side of the equation.
1.25 + -1.25 + z = 0.75 + -1.25

Combine like terms: 1.25 + -1.25 = 0.00
0.00 + z = 0.75 + -1.25
z = 0.75 + -1.25

Combine like terms: 0.75 + -1.25 = -0.5
z = -0.5

Simplifying
z = -0.5

Subproblem 2
z + 1.25 = -0.75

Simplifying
z + 1.25 = -0.75

Reorder the terms:
1.25 + z = -0.75

Solving
1.25 + z = -0.75

Solving for variable 'z'.

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

Add '-1.25' to each side of the equation.
1.25 + -1.25 + z = -0.75 + -1.25

Combine like terms: 1.25 + -1.25 = 0.00
0.00 + z = -0.75 + -1.25
z = -0.75 + -1.25

Combine like terms: -0.75 + -1.25 = -2
z = -2

Simplifying
z = -2

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {-0.5, -2}
Solution
z = {-0.5, -2}