x^2+14x+41=0

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Solution for x^2+14x+41=0 equation: 


Simplifying
x2 + 14x + 41 = 0

Reorder the terms:
41 + 14x + x2 = 0

Solving
41 + 14x + x2 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '-41' to each side of the equation.
41 + 14x + -41 + x2 = 0 + -41

Reorder the terms:
41 + -41 + 14x + x2 = 0 + -41

Combine like terms: 41 + -41 = 0
0 + 14x + x2 = 0 + -41
14x + x2 = 0 + -41

Combine like terms: 0 + -41 = -41
14x + x2 = -41

The x term is 14x.  Take half its coefficient (7).
Square it (49) and add it to both sides.

Add '49' to each side of the equation.
14x + 49 + x2 = -41 + 49

Reorder the terms:
49 + 14x + x2 = -41 + 49

Combine like terms: -41 + 49 = 8
49 + 14x + x2 = 8

Factor a perfect square on the left side:
(x + 7)(x + 7) = 8

Calculate the square root of the right side: 2.828427125

Break this problem into two subproblems by setting 
(x + 7) equal to 2.828427125 and -2.828427125.


Subproblem 1
x + 7 = 2.828427125

Simplifying
x + 7 = 2.828427125

Reorder the terms:
7 + x = 2.828427125

Solving
7 + x = 2.828427125

Solving for variable 'x'.

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

Add '-7' to each side of the equation.
7 + -7 + x = 2.828427125 + -7

Combine like terms: 7 + -7 = 0
0 + x = 2.828427125 + -7
x = 2.828427125 + -7

Combine like terms: 2.828427125 + -7 = -4.171572875
x = -4.171572875

Simplifying
x = -4.171572875


Subproblem 2
x + 7 = -2.828427125

Simplifying
x + 7 = -2.828427125

Reorder the terms:
7 + x = -2.828427125

Solving
7 + x = -2.828427125

Solving for variable 'x'.

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

Add '-7' to each side of the equation.
7 + -7 + x = -2.828427125 + -7

Combine like terms: 7 + -7 = 0
0 + x = -2.828427125 + -7
x = -2.828427125 + -7

Combine like terms: -2.828427125 + -7 = -9.828427125
x = -9.828427125

Simplifying
x = -9.828427125


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-4.171572875, -9.828427125}

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