(6x+1)(6x+1)=7

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Solution for (6x+1)(6x+1)=7 equation: 


Simplifying
(6x + 1)(6x + 1) = 7

Reorder the terms:
(1 + 6x)(6x + 1) = 7

Reorder the terms:
(1 + 6x)(1 + 6x) = 7

Multiply (1 + 6x) * (1 + 6x)
(1(1 + 6x) + 6x * (1 + 6x)) = 7
((1 * 1 + 6x * 1) + 6x * (1 + 6x)) = 7
((1 + 6x) + 6x * (1 + 6x)) = 7
(1 + 6x + (1 * 6x + 6x * 6x)) = 7
(1 + 6x + (6x + 36x2)) = 7

Combine like terms: 6x + 6x = 12x
(1 + 12x + 36x2) = 7

Solving
1 + 12x + 36x2 = 7

Solving for variable 'x'.

Reorder the terms:
1 + -7 + 12x + 36x2 = 7 + -7

Combine like terms: 1 + -7 = -6
-6 + 12x + 36x2 = 7 + -7

Combine like terms: 7 + -7 = 0
-6 + 12x + 36x2 = 0

Factor out the Greatest Common Factor (GCF), '6'.
6(-1 + 2x + 6x2) = 0

Ignore the factor 6.

Subproblem 1
Set the factor '(-1 + 2x + 6x2)' equal to zero and attempt to solve:

Simplifying
-1 + 2x + 6x2 = 0

Solving
-1 + 2x + 6x2 = 0

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

Divide each side by '6'.
-0.1666666667 + 0.3333333333x + x2 = 0

Move the constant term to the right:

Add '0.1666666667' to each side of the equation.
-0.1666666667 + 0.3333333333x + 0.1666666667 + x2 = 0 + 0.1666666667

Reorder the terms:
-0.1666666667 + 0.1666666667 + 0.3333333333x + x2 = 0 + 0.1666666667

Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000
0.0000000000 + 0.3333333333x + x2 = 0 + 0.1666666667
0.3333333333x + x2 = 0 + 0.1666666667

Combine like terms: 0 + 0.1666666667 = 0.1666666667
0.3333333333x + x2 = 0.1666666667

The x term is 0.3333333333x.  Take half its coefficient (0.1666666667).
Square it (0.02777777779) and add it to both sides.

Add '0.02777777779' to each side of the equation.
0.3333333333x + 0.02777777779 + x2 = 0.1666666667 + 0.02777777779

Reorder the terms:
0.02777777779 + 0.3333333333x + x2 = 0.1666666667 + 0.02777777779

Combine like terms: 0.1666666667 + 0.02777777779 = 0.19444444449
0.02777777779 + 0.3333333333x + x2 = 0.19444444449

Factor a perfect square on the left side:
(x + 0.1666666667)(x + 0.1666666667) = 0.19444444449

Calculate the square root of the right side: 0.440958552

Break this problem into two subproblems by setting 
(x + 0.1666666667) equal to 0.440958552 and -0.440958552.


Subproblem 1
x + 0.1666666667 = 0.440958552

Simplifying
x + 0.1666666667 = 0.440958552

Reorder the terms:
0.1666666667 + x = 0.440958552

Solving
0.1666666667 + x = 0.440958552

Solving for variable 'x'.

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

Add '-0.1666666667' to each side of the equation.
0.1666666667 + -0.1666666667 + x = 0.440958552 + -0.1666666667

Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000
0.0000000000 + x = 0.440958552 + -0.1666666667
x = 0.440958552 + -0.1666666667

Combine like terms: 0.440958552 + -0.1666666667 = 0.2742918853
x = 0.2742918853

Simplifying
x = 0.2742918853


Subproblem 2
x + 0.1666666667 = -0.440958552

Simplifying
x + 0.1666666667 = -0.440958552

Reorder the terms:
0.1666666667 + x = -0.440958552

Solving
0.1666666667 + x = -0.440958552

Solving for variable 'x'.

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

Add '-0.1666666667' to each side of the equation.
0.1666666667 + -0.1666666667 + x = -0.440958552 + -0.1666666667

Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000
0.0000000000 + x = -0.440958552 + -0.1666666667
x = -0.440958552 + -0.1666666667

Combine like terms: -0.440958552 + -0.1666666667 = -0.6076252187
x = -0.6076252187

Simplifying
x = -0.6076252187


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.2742918853, -0.6076252187}
Solution
x = {0.2742918853, -0.6076252187}

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