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

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


Simplifying
4x2 + 4x + -1 = 1

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

Solving
-1 + 4x + 4x2 = 1

Solving for variable 'x'.

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

Combine like terms: -1 + -1 = -2
-2 + 4x + 4x2 = 1 + -1

Combine like terms: 1 + -1 = 0
-2 + 4x + 4x2 = 0

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

Ignore the factor 2.

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

Simplifying
-1 + 2x + 2x2 = 0

Solving
-1 + 2x + 2x2 = 0

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

Divide each side by '2'.
-0.5 + x + x2 = 0

Move the constant term to the right:

Add '0.5' to each side of the equation.
-0.5 + x + 0.5 + x2 = 0 + 0.5

Reorder the terms:
-0.5 + 0.5 + x + x2 = 0 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + x + x2 = 0 + 0.5
x + x2 = 0 + 0.5

Combine like terms: 0 + 0.5 = 0.5
x + x2 = 0.5

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.
 + 0.25 + x2 = 0.5 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + x2 = 0.5 + 0.25

Combine like terms: 0.5 + 0.25 = 0.75
1.25 + x2 = 0.75

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 0.75

Calculate the square root of the right side: 0.866025404

Break this problem into two subproblems by setting 
(x + 0.5) equal to 0.866025404 and -0.866025404.


Subproblem 1
x + 0.5 = 0.866025404

Simplifying
x + 0.5 = 0.866025404

Reorder the terms:
0.5 + x = 0.866025404

Solving
0.5 + x = 0.866025404

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 = 0.866025404 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 0.866025404 + -0.5
x = 0.866025404 + -0.5

Combine like terms: 0.866025404 + -0.5 = 0.366025404
x = 0.366025404

Simplifying
x = 0.366025404


Subproblem 2
x + 0.5 = -0.866025404

Simplifying
x + 0.5 = -0.866025404

Reorder the terms:
0.5 + x = -0.866025404

Solving
0.5 + x = -0.866025404

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 = -0.866025404 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -0.866025404 + -0.5
x = -0.866025404 + -0.5

Combine like terms: -0.866025404 + -0.5 = -1.366025404
x = -1.366025404

Simplifying
x = -1.366025404


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.366025404, -1.366025404}
Solution
x = {0.366025404, -1.366025404}

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