-21z^6+49z^4-9z^2=0

Solution for -21z^6+49z^4-9z^2=0 equation:

Simplifying
-21z6 + 49z4 + -9z2 = 0

Reorder the terms:
-9z2 + 49z4 + -21z6 = 0

Solving
-9z2 + 49z4 + -21z6 = 0

Solving for variable 'z'.

Factor out the Greatest Common Factor (GCF), 'z2'.
z2(-9 + 49z2 + -21z4) = 0

Subproblem 1
Set the factor 'z2' equal to zero and attempt to solve:

Simplifying
z2 = 0

Solving
z2 = 0

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

Simplifying
z2 = 0

Take the square root of each side:
z = {0}
Subproblem 2
Set the factor '(-9 + 49z2 + -21z4)' equal to zero and attempt to solve:

Simplifying
-9 + 49z2 + -21z4 = 0

Solving
-9 + 49z2 + -21z4 = 0

Begin completing the square.  Divide all terms by
-21 the coefficient of the squared term: 

Divide each side by '-21'.
0.4285714286 + -2.333333333z2 + z4 = 0

Move the constant term to the right:

Add '-0.4285714286' to each side of the equation.
0.4285714286 + -2.333333333z2 + -0.4285714286 + z4 = 0 + -0.4285714286

Reorder the terms:
0.4285714286 + -0.4285714286 + -2.333333333z2 + z4 = 0 + -0.4285714286

Combine like terms: 0.4285714286 + -0.4285714286 = 0.0000000000
0.0000000000 + -2.333333333z2 + z4 = 0 + -0.4285714286
-2.333333333z2 + z4 = 0 + -0.4285714286

Combine like terms: 0 + -0.4285714286 = -0.4285714286
-2.333333333z2 + z4 = -0.4285714286

The z term is -2.333333333z2.  Take half its coefficient (-1.166666667).
Square it (1.361111112) and add it to both sides.

Add '1.361111112' to each side of the equation.
-2.333333333z2 + 1.361111112 + z4 = -0.4285714286 + 1.361111112

Reorder the terms:
1.361111112 + -2.333333333z2 + z4 = -0.4285714286 + 1.361111112

Combine like terms: -0.4285714286 + 1.361111112 = 0.9325396834
1.361111112 + -2.333333333z2 + z4 = 0.9325396834

Factor a perfect square on the left side:
(z2 + -1.166666667)(z2 + -1.166666667) = 0.9325396834

Calculate the square root of the right side: 0.965680943

Break this problem into two subproblems by setting 
(z2 + -1.166666667) equal to 0.965680943 and -0.965680943.

Subproblem 1
z2 + -1.166666667 = 0.965680943

Simplifying
z2 + -1.166666667 = 0.965680943

Reorder the terms:
-1.166666667 + z2 = 0.965680943

Solving
-1.166666667 + z2 = 0.965680943

Solving for variable 'z'.

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

Add '1.166666667' to each side of the equation.
-1.166666667 + 1.166666667 + z2 = 0.965680943 + 1.166666667

Combine like terms: -1.166666667 + 1.166666667 = 0.000000000
0.000000000 + z2 = 0.965680943 + 1.166666667
z2 = 0.965680943 + 1.166666667

Combine like terms: 0.965680943 + 1.166666667 = 2.13234761
z2 = 2.13234761

Simplifying
z2 = 2.13234761

Take the square root of each side:
z = {-1.460256008, 1.460256008}

Subproblem 2
z2 + -1.166666667 = -0.965680943

Simplifying
z2 + -1.166666667 = -0.965680943

Reorder the terms:
-1.166666667 + z2 = -0.965680943

Solving
-1.166666667 + z2 = -0.965680943

Solving for variable 'z'.

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

Add '1.166666667' to each side of the equation.
-1.166666667 + 1.166666667 + z2 = -0.965680943 + 1.166666667

Combine like terms: -1.166666667 + 1.166666667 = 0.000000000
0.000000000 + z2 = -0.965680943 + 1.166666667
z2 = -0.965680943 + 1.166666667

Combine like terms: -0.965680943 + 1.166666667 = 0.200985724
z2 = 0.200985724

Simplifying
z2 = 0.200985724

Take the square root of each side:
z = {-0.448314314, 0.448314314}

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {-1.460256008, 1.460256008, -0.448314314, 0.448314314}
Solution
z = {0, -1.460256008, 1.460256008, -0.448314314, 0.448314314}