0=x^2+6x+3

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Solution for 0=x^2+6x+3 equation: 


Simplifying
0 = x2 + 6x + 3

Reorder the terms:
0 = 3 + 6x + x2

Solving
0 = 3 + 6x + x2

Solving for variable 'x'.

Combine like terms: 0 + -3 = -3
-3 + -6x + -1x2 = 3 + 6x + x2 + -3 + -6x + -1x2

Reorder the terms:
-3 + -6x + -1x2 = 3 + -3 + 6x + -6x + x2 + -1x2

Combine like terms: 3 + -3 = 0
-3 + -6x + -1x2 = 0 + 6x + -6x + x2 + -1x2
-3 + -6x + -1x2 = 6x + -6x + x2 + -1x2

Combine like terms: 6x + -6x = 0
-3 + -6x + -1x2 = 0 + x2 + -1x2
-3 + -6x + -1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
-3 + -6x + -1x2 = 0

Factor out the Greatest Common Factor (GCF), '-1'.
-1(3 + 6x + x2) = 0

Ignore the factor -1.

Subproblem 1
Set the factor '(3 + 6x + x2)' equal to zero and attempt to solve:

Simplifying
3 + 6x + x2 = 0

Solving
3 + 6x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-3' to each side of the equation.
3 + 6x + -3 + x2 = 0 + -3

Reorder the terms:
3 + -3 + 6x + x2 = 0 + -3

Combine like terms: 3 + -3 = 0
0 + 6x + x2 = 0 + -3
6x + x2 = 0 + -3

Combine like terms: 0 + -3 = -3
6x + x2 = -3

The x term is 6x.  Take half its coefficient (3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
6x + 9 + x2 = -3 + 9

Reorder the terms:
9 + 6x + x2 = -3 + 9

Combine like terms: -3 + 9 = 6
9 + 6x + x2 = 6

Factor a perfect square on the left side:
(x + 3)(x + 3) = 6

Calculate the square root of the right side: 2.449489743

Break this problem into two subproblems by setting 
(x + 3) equal to 2.449489743 and -2.449489743.


Subproblem 1
x + 3 = 2.449489743

Simplifying
x + 3 = 2.449489743

Reorder the terms:
3 + x = 2.449489743

Solving
3 + x = 2.449489743

Solving for variable 'x'.

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

Add '-3' to each side of the equation.
3 + -3 + x = 2.449489743 + -3

Combine like terms: 3 + -3 = 0
0 + x = 2.449489743 + -3
x = 2.449489743 + -3

Combine like terms: 2.449489743 + -3 = -0.550510257
x = -0.550510257

Simplifying
x = -0.550510257


Subproblem 2
x + 3 = -2.449489743

Simplifying
x + 3 = -2.449489743

Reorder the terms:
3 + x = -2.449489743

Solving
3 + x = -2.449489743

Solving for variable 'x'.

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

Add '-3' to each side of the equation.
3 + -3 + x = -2.449489743 + -3

Combine like terms: 3 + -3 = 0
0 + x = -2.449489743 + -3
x = -2.449489743 + -3

Combine like terms: -2.449489743 + -3 = -5.449489743
x = -5.449489743

Simplifying
x = -5.449489743


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.550510257, -5.449489743}
Solution
x = {-0.550510257, -5.449489743}

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