18x^2z^2-4xz^2-12z^2=0

Solution for 18x^2z^2-4xz^2-12z^2=0 equation:

Simplifying
18x2z2 + -4xz2 + -12z2 = 0

Reorder the terms:
-4xz2 + 18x2z2 + -12z2 = 0

Solving
-4xz2 + 18x2z2 + -12z2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '2z2'.
2z2(-2x + 9x2 + -6) = 0

Ignore the factor 2.
Subproblem 1
Set the factor 'z2' equal to zero and attempt to solve:

Simplifying
z2 = 0

Solving
z2 = 0

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

Add '-1z2' to each side of the equation.
z2 + -1z2 = 0 + -1z2
Remove the zero:
0 = -1z2

Simplifying
0 = -1z2

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 '(-2x + 9x2 + -6)' equal to zero and attempt to solve:

Simplifying
-2x + 9x2 + -6 = 0

Reorder the terms:
-6 + -2x + 9x2 = 0

Solving
-6 + -2x + 9x2 = 0

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

Divide each side by '9'.
-0.6666666667 + -0.2222222222x + x2 = 0

Move the constant term to the right:

Add '0.6666666667' to each side of the equation.
-0.6666666667 + -0.2222222222x + 0.6666666667 + x2 = 0 + 0.6666666667

Reorder the terms:
-0.6666666667 + 0.6666666667 + -0.2222222222x + x2 = 0 + 0.6666666667

Combine like terms: -0.6666666667 + 0.6666666667 = 0.0000000000
0.0000000000 + -0.2222222222x + x2 = 0 + 0.6666666667
-0.2222222222x + x2 = 0 + 0.6666666667

Combine like terms: 0 + 0.6666666667 = 0.6666666667
-0.2222222222x + x2 = 0.6666666667

The x term is -0.2222222222x.  Take half its coefficient (-0.1111111111).
Square it (0.01234567901) and add it to both sides.

Add '0.01234567901' to each side of the equation.
-0.2222222222x + 0.01234567901 + x2 = 0.6666666667 + 0.01234567901

Reorder the terms:
0.01234567901 + -0.2222222222x + x2 = 0.6666666667 + 0.01234567901

Combine like terms: 0.6666666667 + 0.01234567901 = 0.67901234571
0.01234567901 + -0.2222222222x + x2 = 0.67901234571

Factor a perfect square on the left side:
(x + -0.1111111111)(x + -0.1111111111) = 0.67901234571

Calculate the square root of the right side: 0.824022054

Break this problem into two subproblems by setting 
(x + -0.1111111111) equal to 0.824022054 and -0.824022054.

Subproblem 1
x + -0.1111111111 = 0.824022054

Simplifying
x + -0.1111111111 = 0.824022054

Reorder the terms:
-0.1111111111 + x = 0.824022054

Solving
-0.1111111111 + x = 0.824022054

Solving for variable 'x'.

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

Add '0.1111111111' to each side of the equation.
-0.1111111111 + 0.1111111111 + x = 0.824022054 + 0.1111111111

Combine like terms: -0.1111111111 + 0.1111111111 = 0.0000000000
0.0000000000 + x = 0.824022054 + 0.1111111111
x = 0.824022054 + 0.1111111111

Combine like terms: 0.824022054 + 0.1111111111 = 0.9351331651
x = 0.9351331651

Simplifying
x = 0.9351331651

Subproblem 2
x + -0.1111111111 = -0.824022054

Simplifying
x + -0.1111111111 = -0.824022054

Reorder the terms:
-0.1111111111 + x = -0.824022054

Solving
-0.1111111111 + x = -0.824022054

Solving for variable 'x'.

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

Add '0.1111111111' to each side of the equation.
-0.1111111111 + 0.1111111111 + x = -0.824022054 + 0.1111111111

Combine like terms: -0.1111111111 + 0.1111111111 = 0.0000000000
0.0000000000 + x = -0.824022054 + 0.1111111111
x = -0.824022054 + 0.1111111111

Combine like terms: -0.824022054 + 0.1111111111 = -0.7129109429
x = -0.7129109429

Simplifying
x = -0.7129109429

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.9351331651, -0.7129109429}
Solution
x = {0.9351331651, -0.7129109429}