8x^2+18x+6=0

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Solution for 8x^2+18x+6=0 equation: 


Simplifying
8x2 + 18x + 6 = 0

Reorder the terms:
6 + 18x + 8x2 = 0

Solving
6 + 18x + 8x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '2'.
2(3 + 9x + 4x2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(3 + 9x + 4x2)' equal to zero and attempt to solve:

Simplifying
3 + 9x + 4x2 = 0

Solving
3 + 9x + 4x2 = 0

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

Divide each side by '4'.
0.75 + 2.25x + x2 = 0

Move the constant term to the right:

Add '-0.75' to each side of the equation.
0.75 + 2.25x + -0.75 + x2 = 0 + -0.75

Reorder the terms:
0.75 + -0.75 + 2.25x + x2 = 0 + -0.75

Combine like terms: 0.75 + -0.75 = 0.00
0.00 + 2.25x + x2 = 0 + -0.75
2.25x + x2 = 0 + -0.75

Combine like terms: 0 + -0.75 = -0.75
2.25x + x2 = -0.75

The x term is 2.25x.  Take half its coefficient (1.125).
Square it (1.265625) and add it to both sides.

Add '1.265625' to each side of the equation.
2.25x + 1.265625 + x2 = -0.75 + 1.265625

Reorder the terms:
1.265625 + 2.25x + x2 = -0.75 + 1.265625

Combine like terms: -0.75 + 1.265625 = 0.515625
1.265625 + 2.25x + x2 = 0.515625

Factor a perfect square on the left side:
(x + 1.125)(x + 1.125) = 0.515625

Calculate the square root of the right side: 0.718070331

Break this problem into two subproblems by setting 
(x + 1.125) equal to 0.718070331 and -0.718070331.


Subproblem 1
x + 1.125 = 0.718070331

Simplifying
x + 1.125 = 0.718070331

Reorder the terms:
1.125 + x = 0.718070331

Solving
1.125 + x = 0.718070331

Solving for variable 'x'.

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

Add '-1.125' to each side of the equation.
1.125 + -1.125 + x = 0.718070331 + -1.125

Combine like terms: 1.125 + -1.125 = 0.000
0.000 + x = 0.718070331 + -1.125
x = 0.718070331 + -1.125

Combine like terms: 0.718070331 + -1.125 = -0.406929669
x = -0.406929669

Simplifying
x = -0.406929669


Subproblem 2
x + 1.125 = -0.718070331

Simplifying
x + 1.125 = -0.718070331

Reorder the terms:
1.125 + x = -0.718070331

Solving
1.125 + x = -0.718070331

Solving for variable 'x'.

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

Add '-1.125' to each side of the equation.
1.125 + -1.125 + x = -0.718070331 + -1.125

Combine like terms: 1.125 + -1.125 = 0.000
0.000 + x = -0.718070331 + -1.125
x = -0.718070331 + -1.125

Combine like terms: -0.718070331 + -1.125 = -1.843070331
x = -1.843070331

Simplifying
x = -1.843070331


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.406929669, -1.843070331}
Solution
x = {-0.406929669, -1.843070331}

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