7x(6x+1)=511

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Solution for 7x(6x+1)=511 equation: 


Simplifying
7x(6x + 1) = 511

Reorder the terms:
7x(1 + 6x) = 511
(1 * 7x + 6x * 7x) = 511
(7x + 42x2) = 511

Solving
7x + 42x2 = 511

Solving for variable 'x'.

Reorder the terms:
-511 + 7x + 42x2 = 511 + -511

Combine like terms: 511 + -511 = 0
-511 + 7x + 42x2 = 0

Factor out the Greatest Common Factor (GCF), '7'.
7(-73 + x + 6x2) = 0

Ignore the factor 7.

Subproblem 1
Set the factor '(-73 + x + 6x2)' equal to zero and attempt to solve:

Simplifying
-73 + x + 6x2 = 0

Solving
-73 + x + 6x2 = 0

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

Divide each side by '6'.
-12.16666667 + 0.1666666667x + x2 = 0

Move the constant term to the right:

Add '12.16666667' to each side of the equation.
-12.16666667 + 0.1666666667x + 12.16666667 + x2 = 0 + 12.16666667

Reorder the terms:
-12.16666667 + 12.16666667 + 0.1666666667x + x2 = 0 + 12.16666667

Combine like terms: -12.16666667 + 12.16666667 = 0.00000000
0.00000000 + 0.1666666667x + x2 = 0 + 12.16666667
0.1666666667x + x2 = 0 + 12.16666667

Combine like terms: 0 + 12.16666667 = 12.16666667
0.1666666667x + x2 = 12.16666667

The x term is x.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
0.1666666667x + 0.25 + x2 = 12.16666667 + 0.25

Reorder the terms:
0.25 + 0.1666666667x + x2 = 12.16666667 + 0.25

Combine like terms: 12.16666667 + 0.25 = 12.41666667
0.25 + 0.1666666667x + x2 = 12.41666667

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 12.41666667

Calculate the square root of the right side: 3.523729086

Break this problem into two subproblems by setting 
(x + 0.5) equal to 3.523729086 and -3.523729086.


Subproblem 1
x + 0.5 = 3.523729086

Simplifying
x + 0.5 = 3.523729086

Reorder the terms:
0.5 + x = 3.523729086

Solving
0.5 + x = 3.523729086

Solving for variable 'x'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = 3.523729086 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 3.523729086 + -0.5
x = 3.523729086 + -0.5

Combine like terms: 3.523729086 + -0.5 = 3.023729086
x = 3.023729086

Simplifying
x = 3.023729086


Subproblem 2
x + 0.5 = -3.523729086

Simplifying
x + 0.5 = -3.523729086

Reorder the terms:
0.5 + x = -3.523729086

Solving
0.5 + x = -3.523729086

Solving for variable 'x'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = -3.523729086 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -3.523729086 + -0.5
x = -3.523729086 + -0.5

Combine like terms: -3.523729086 + -0.5 = -4.023729086
x = -4.023729086

Simplifying
x = -4.023729086


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3.023729086, -4.023729086}
Solution
x = {3.023729086, -4.023729086}

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