81x^2+144x+54=0

Solution for 81x^2+144x+54=0 equation:

Simplifying
81x2 + 144x + 54 = 0

Reorder the terms:
54 + 144x + 81x2 = 0

Solving
54 + 144x + 81x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '9'.
9(6 + 16x + 9x2) = 0

Ignore the factor 9.
Subproblem 1
Set the factor '(6 + 16x + 9x2)' equal to zero and attempt to solve:

Simplifying
6 + 16x + 9x2 = 0

Solving
6 + 16x + 9x2 = 0

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

Divide each side by '9'.
0.6666666667 + 1.777777778x + x2 = 0

Move the constant term to the right:

Add '-0.6666666667' to each side of the equation.
0.6666666667 + 1.777777778x + -0.6666666667 + x2 = 0 + -0.6666666667

Reorder the terms:
0.6666666667 + -0.6666666667 + 1.777777778x + x2 = 0 + -0.6666666667

Combine like terms: 0.6666666667 + -0.6666666667 = 0.0000000000
0.0000000000 + 1.777777778x + x2 = 0 + -0.6666666667
1.777777778x + x2 = 0 + -0.6666666667

Combine like terms: 0 + -0.6666666667 = -0.6666666667
1.777777778x + x2 = -0.6666666667

The x term is 1.777777778x.  Take half its coefficient (0.888888889).
Square it (0.7901234570) and add it to both sides.

Add '0.7901234570' to each side of the equation.
1.777777778x + 0.7901234570 + x2 = -0.6666666667 + 0.7901234570

Reorder the terms:
0.7901234570 + 1.777777778x + x2 = -0.6666666667 + 0.7901234570

Combine like terms: -0.6666666667 + 0.7901234570 = 0.1234567903
0.7901234570 + 1.777777778x + x2 = 0.1234567903

Factor a perfect square on the left side:
(x + 0.888888889)(x + 0.888888889) = 0.1234567903

Calculate the square root of the right side: 0.351364185

Break this problem into two subproblems by setting 
(x + 0.888888889) equal to 0.351364185 and -0.351364185.

Subproblem 1
x + 0.888888889 = 0.351364185

Simplifying
x + 0.888888889 = 0.351364185

Reorder the terms:
0.888888889 + x = 0.351364185

Solving
0.888888889 + x = 0.351364185

Solving for variable 'x'.

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

Add '-0.888888889' to each side of the equation.
0.888888889 + -0.888888889 + x = 0.351364185 + -0.888888889

Combine like terms: 0.888888889 + -0.888888889 = 0.000000000
0.000000000 + x = 0.351364185 + -0.888888889
x = 0.351364185 + -0.888888889

Combine like terms: 0.351364185 + -0.888888889 = -0.537524704
x = -0.537524704

Simplifying
x = -0.537524704

Subproblem 2
x + 0.888888889 = -0.351364185

Simplifying
x + 0.888888889 = -0.351364185

Reorder the terms:
0.888888889 + x = -0.351364185

Solving
0.888888889 + x = -0.351364185

Solving for variable 'x'.

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

Add '-0.888888889' to each side of the equation.
0.888888889 + -0.888888889 + x = -0.351364185 + -0.888888889

Combine like terms: 0.888888889 + -0.888888889 = 0.000000000
0.000000000 + x = -0.351364185 + -0.888888889
x = -0.351364185 + -0.888888889

Combine like terms: -0.351364185 + -0.888888889 = -1.240253074
x = -1.240253074

Simplifying
x = -1.240253074

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.537524704, -1.240253074}
Solution
x = {-0.537524704, -1.240253074}