0=x^2+4x+2

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Solution for 0=x^2+4x+2 equation: 


Simplifying
0 = x2 + 4x + 2

Reorder the terms:
0 = 2 + 4x + x2

Solving
0 = 2 + 4x + x2

Solving for variable 'x'.

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

Reorder the terms:
-2 + -4x + -1x2 = 2 + -2 + 4x + -4x + x2 + -1x2

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

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

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

Factor out the Greatest Common Factor (GCF), '-1'.
-1(2 + 4x + x2) = 0

Ignore the factor -1.

Subproblem 1
Set the factor '(2 + 4x + x2)' equal to zero and attempt to solve:

Simplifying
2 + 4x + x2 = 0

Solving
2 + 4x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-2' to each side of the equation.
2 + 4x + -2 + x2 = 0 + -2

Reorder the terms:
2 + -2 + 4x + x2 = 0 + -2

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

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

The x term is 4x.  Take half its coefficient (2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
4x + 4 + x2 = -2 + 4

Reorder the terms:
4 + 4x + x2 = -2 + 4

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

Factor a perfect square on the left side:
(x + 2)(x + 2) = 2

Calculate the square root of the right side: 1.414213562

Break this problem into two subproblems by setting 
(x + 2) equal to 1.414213562 and -1.414213562.


Subproblem 1
x + 2 = 1.414213562

Simplifying
x + 2 = 1.414213562

Reorder the terms:
2 + x = 1.414213562

Solving
2 + x = 1.414213562

Solving for variable 'x'.

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

Add '-2' to each side of the equation.
2 + -2 + x = 1.414213562 + -2

Combine like terms: 2 + -2 = 0
0 + x = 1.414213562 + -2
x = 1.414213562 + -2

Combine like terms: 1.414213562 + -2 = -0.585786438
x = -0.585786438

Simplifying
x = -0.585786438


Subproblem 2
x + 2 = -1.414213562

Simplifying
x + 2 = -1.414213562

Reorder the terms:
2 + x = -1.414213562

Solving
2 + x = -1.414213562

Solving for variable 'x'.

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

Add '-2' to each side of the equation.
2 + -2 + x = -1.414213562 + -2

Combine like terms: 2 + -2 = 0
0 + x = -1.414213562 + -2
x = -1.414213562 + -2

Combine like terms: -1.414213562 + -2 = -3.414213562
x = -3.414213562

Simplifying
x = -3.414213562


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.585786438, -3.414213562}
Solution
x = {-0.585786438, -3.414213562}

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