x^2+x=4x^2-4x+1

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Solution for x^2+x=4x^2-4x+1 equation: 


Simplifying
x2 + x = 4x2 + -4x + 1

Reorder the terms:
x + x2 = 4x2 + -4x + 1

Reorder the terms:
x + x2 = 1 + -4x + 4x2

Solving
x + x2 = 1 + -4x + 4x2

Solving for variable 'x'.

Reorder the terms:
-1 + x + 4x + x2 + -4x2 = 1 + -4x + 4x2 + -1 + 4x + -4x2

Combine like terms: x + 4x = 5x
-1 + 5x + x2 + -4x2 = 1 + -4x + 4x2 + -1 + 4x + -4x2

Combine like terms: x2 + -4x2 = -3x2
-1 + 5x + -3x2 = 1 + -4x + 4x2 + -1 + 4x + -4x2

Reorder the terms:
-1 + 5x + -3x2 = 1 + -1 + -4x + 4x + 4x2 + -4x2

Combine like terms: 1 + -1 = 0
-1 + 5x + -3x2 = 0 + -4x + 4x + 4x2 + -4x2
-1 + 5x + -3x2 = -4x + 4x + 4x2 + -4x2

Combine like terms: -4x + 4x = 0
-1 + 5x + -3x2 = 0 + 4x2 + -4x2
-1 + 5x + -3x2 = 4x2 + -4x2

Combine like terms: 4x2 + -4x2 = 0
-1 + 5x + -3x2 = 0

Begin completing the square.  Divide all terms by
-3 the coefficient of the squared term: 

Divide each side by '-3'.
0.3333333333 + -1.666666667x + x2 = 0

Move the constant term to the right:

Add '-0.3333333333' to each side of the equation.
0.3333333333 + -1.666666667x + -0.3333333333 + x2 = 0 + -0.3333333333

Reorder the terms:
0.3333333333 + -0.3333333333 + -1.666666667x + x2 = 0 + -0.3333333333

Combine like terms: 0.3333333333 + -0.3333333333 = 0.0000000000
0.0000000000 + -1.666666667x + x2 = 0 + -0.3333333333
-1.666666667x + x2 = 0 + -0.3333333333

Combine like terms: 0 + -0.3333333333 = -0.3333333333
-1.666666667x + x2 = -0.3333333333

The x term is -1.666666667x.  Take half its coefficient (-0.8333333335).
Square it (0.6944444447) and add it to both sides.

Add '0.6944444447' to each side of the equation.
-1.666666667x + 0.6944444447 + x2 = -0.3333333333 + 0.6944444447

Reorder the terms:
0.6944444447 + -1.666666667x + x2 = -0.3333333333 + 0.6944444447

Combine like terms: -0.3333333333 + 0.6944444447 = 0.3611111114
0.6944444447 + -1.666666667x + x2 = 0.3611111114

Factor a perfect square on the left side:
(x + -0.8333333335)(x + -0.8333333335) = 0.3611111114

Calculate the square root of the right side: 0.600925213

Break this problem into two subproblems by setting 
(x + -0.8333333335) equal to 0.600925213 and -0.600925213.


Subproblem 1
x + -0.8333333335 = 0.600925213

Simplifying
x + -0.8333333335 = 0.600925213

Reorder the terms:
-0.8333333335 + x = 0.600925213

Solving
-0.8333333335 + x = 0.600925213

Solving for variable 'x'.

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

Add '0.8333333335' to each side of the equation.
-0.8333333335 + 0.8333333335 + x = 0.600925213 + 0.8333333335

Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000
0.0000000000 + x = 0.600925213 + 0.8333333335
x = 0.600925213 + 0.8333333335

Combine like terms: 0.600925213 + 0.8333333335 = 1.4342585465
x = 1.4342585465

Simplifying
x = 1.4342585465


Subproblem 2
x + -0.8333333335 = -0.600925213

Simplifying
x + -0.8333333335 = -0.600925213

Reorder the terms:
-0.8333333335 + x = -0.600925213

Solving
-0.8333333335 + x = -0.600925213

Solving for variable 'x'.

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

Add '0.8333333335' to each side of the equation.
-0.8333333335 + 0.8333333335 + x = -0.600925213 + 0.8333333335

Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000
0.0000000000 + x = -0.600925213 + 0.8333333335
x = -0.600925213 + 0.8333333335

Combine like terms: -0.600925213 + 0.8333333335 = 0.2324081205
x = 0.2324081205

Simplifying
x = 0.2324081205


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.4342585465, 0.2324081205}

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