4x(3x+7)=1

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Solution for 4x(3x+7)=1 equation: 


Simplifying
4x(3x + 7) = 1

Reorder the terms:
4x(7 + 3x) = 1
(7 * 4x + 3x * 4x) = 1
(28x + 12x2) = 1

Solving
28x + 12x2 = 1

Solving for variable 'x'.

Reorder the terms:
-1 + 28x + 12x2 = 1 + -1

Combine like terms: 1 + -1 = 0
-1 + 28x + 12x2 = 0

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

Divide each side by '12'.
-0.08333333333 + 2.333333333x + x2 = 0

Move the constant term to the right:

Add '0.08333333333' to each side of the equation.
-0.08333333333 + 2.333333333x + 0.08333333333 + x2 = 0 + 0.08333333333

Reorder the terms:
-0.08333333333 + 0.08333333333 + 2.333333333x + x2 = 0 + 0.08333333333

Combine like terms: -0.08333333333 + 0.08333333333 = 0.00000000000
0.00000000000 + 2.333333333x + x2 = 0 + 0.08333333333
2.333333333x + x2 = 0 + 0.08333333333

Combine like terms: 0 + 0.08333333333 = 0.08333333333
2.333333333x + x2 = 0.08333333333

The x term is 2.333333333x.  Take half its coefficient (1.166666667).
Square it (1.361111112) and add it to both sides.

Add '1.361111112' to each side of the equation.
2.333333333x + 1.361111112 + x2 = 0.08333333333 + 1.361111112

Reorder the terms:
1.361111112 + 2.333333333x + x2 = 0.08333333333 + 1.361111112

Combine like terms: 0.08333333333 + 1.361111112 = 1.44444444533
1.361111112 + 2.333333333x + x2 = 1.44444444533

Factor a perfect square on the left side:
(x + 1.166666667)(x + 1.166666667) = 1.44444444533

Calculate the square root of the right side: 1.201850426

Break this problem into two subproblems by setting 
(x + 1.166666667) equal to 1.201850426 and -1.201850426.


Subproblem 1
x + 1.166666667 = 1.201850426

Simplifying
x + 1.166666667 = 1.201850426

Reorder the terms:
1.166666667 + x = 1.201850426

Solving
1.166666667 + x = 1.201850426

Solving for variable 'x'.

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

Add '-1.166666667' to each side of the equation.
1.166666667 + -1.166666667 + x = 1.201850426 + -1.166666667

Combine like terms: 1.166666667 + -1.166666667 = 0.000000000
0.000000000 + x = 1.201850426 + -1.166666667
x = 1.201850426 + -1.166666667

Combine like terms: 1.201850426 + -1.166666667 = 0.035183759
x = 0.035183759

Simplifying
x = 0.035183759


Subproblem 2
x + 1.166666667 = -1.201850426

Simplifying
x + 1.166666667 = -1.201850426

Reorder the terms:
1.166666667 + x = -1.201850426

Solving
1.166666667 + x = -1.201850426

Solving for variable 'x'.

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

Add '-1.166666667' to each side of the equation.
1.166666667 + -1.166666667 + x = -1.201850426 + -1.166666667

Combine like terms: 1.166666667 + -1.166666667 = 0.000000000
0.000000000 + x = -1.201850426 + -1.166666667
x = -1.201850426 + -1.166666667

Combine like terms: -1.201850426 + -1.166666667 = -2.368517093
x = -2.368517093

Simplifying
x = -2.368517093


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.035183759, -2.368517093}

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