4x^2-18x-6=0

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Solution for 4x^2-18x-6=0 equation: 


Simplifying
4x2 + -18x + -6 = 0

Reorder the terms:
-6 + -18x + 4x2 = 0

Solving
-6 + -18x + 4x2 = 0

Solving for variable 'x'.

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

Ignore the factor 2.

Subproblem 1
Set the factor '(-3 + -9x + 2x2)' equal to zero and attempt to solve:

Simplifying
-3 + -9x + 2x2 = 0

Solving
-3 + -9x + 2x2 = 0

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

Divide each side by '2'.
-1.5 + -4.5x + x2 = 0

Move the constant term to the right:

Add '1.5' to each side of the equation.
-1.5 + -4.5x + 1.5 + x2 = 0 + 1.5

Reorder the terms:
-1.5 + 1.5 + -4.5x + x2 = 0 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + -4.5x + x2 = 0 + 1.5
-4.5x + x2 = 0 + 1.5

Combine like terms: 0 + 1.5 = 1.5
-4.5x + x2 = 1.5

The x term is -4.5x.  Take half its coefficient (-2.25).
Square it (5.0625) and add it to both sides.

Add '5.0625' to each side of the equation.
-4.5x + 5.0625 + x2 = 1.5 + 5.0625

Reorder the terms:
5.0625 + -4.5x + x2 = 1.5 + 5.0625

Combine like terms: 1.5 + 5.0625 = 6.5625
5.0625 + -4.5x + x2 = 6.5625

Factor a perfect square on the left side:
(x + -2.25)(x + -2.25) = 6.5625

Calculate the square root of the right side: 2.561737691

Break this problem into two subproblems by setting 
(x + -2.25) equal to 2.561737691 and -2.561737691.


Subproblem 1
x + -2.25 = 2.561737691

Simplifying
x + -2.25 = 2.561737691

Reorder the terms:
-2.25 + x = 2.561737691

Solving
-2.25 + x = 2.561737691

Solving for variable 'x'.

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

Add '2.25' to each side of the equation.
-2.25 + 2.25 + x = 2.561737691 + 2.25

Combine like terms: -2.25 + 2.25 = 0.00
0.00 + x = 2.561737691 + 2.25
x = 2.561737691 + 2.25

Combine like terms: 2.561737691 + 2.25 = 4.811737691
x = 4.811737691

Simplifying
x = 4.811737691


Subproblem 2
x + -2.25 = -2.561737691

Simplifying
x + -2.25 = -2.561737691

Reorder the terms:
-2.25 + x = -2.561737691

Solving
-2.25 + x = -2.561737691

Solving for variable 'x'.

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

Add '2.25' to each side of the equation.
-2.25 + 2.25 + x = -2.561737691 + 2.25

Combine like terms: -2.25 + 2.25 = 0.00
0.00 + x = -2.561737691 + 2.25
x = -2.561737691 + 2.25

Combine like terms: -2.561737691 + 2.25 = -0.311737691
x = -0.311737691

Simplifying
x = -0.311737691


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {4.811737691, -0.311737691}
Solution
x = {4.811737691, -0.311737691}

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