x^2+22x+52=0

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Solution for x^2+22x+52=0 equation: 


Simplifying
x2 + 22x + 52 = 0

Reorder the terms:
52 + 22x + x2 = 0

Solving
52 + 22x + x2 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '-52' to each side of the equation.
52 + 22x + -52 + x2 = 0 + -52

Reorder the terms:
52 + -52 + 22x + x2 = 0 + -52

Combine like terms: 52 + -52 = 0
0 + 22x + x2 = 0 + -52
22x + x2 = 0 + -52

Combine like terms: 0 + -52 = -52
22x + x2 = -52

The x term is 22x.  Take half its coefficient (11).
Square it (121) and add it to both sides.

Add '121' to each side of the equation.
22x + 121 + x2 = -52 + 121

Reorder the terms:
121 + 22x + x2 = -52 + 121

Combine like terms: -52 + 121 = 69
121 + 22x + x2 = 69

Factor a perfect square on the left side:
(x + 11)(x + 11) = 69

Calculate the square root of the right side: 8.306623863

Break this problem into two subproblems by setting 
(x + 11) equal to 8.306623863 and -8.306623863.


Subproblem 1
x + 11 = 8.306623863

Simplifying
x + 11 = 8.306623863

Reorder the terms:
11 + x = 8.306623863

Solving
11 + x = 8.306623863

Solving for variable 'x'.

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

Add '-11' to each side of the equation.
11 + -11 + x = 8.306623863 + -11

Combine like terms: 11 + -11 = 0
0 + x = 8.306623863 + -11
x = 8.306623863 + -11

Combine like terms: 8.306623863 + -11 = -2.693376137
x = -2.693376137

Simplifying
x = -2.693376137


Subproblem 2
x + 11 = -8.306623863

Simplifying
x + 11 = -8.306623863

Reorder the terms:
11 + x = -8.306623863

Solving
11 + x = -8.306623863

Solving for variable 'x'.

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

Add '-11' to each side of the equation.
11 + -11 + x = -8.306623863 + -11

Combine like terms: 11 + -11 = 0
0 + x = -8.306623863 + -11
x = -8.306623863 + -11

Combine like terms: -8.306623863 + -11 = -19.306623863
x = -19.306623863

Simplifying
x = -19.306623863


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-2.693376137, -19.306623863}

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