0=x^2+10x+22

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Solution for 0=x^2+10x+22 equation: 


Simplifying
0 = x2 + 10x + 22

Reorder the terms:
0 = 22 + 10x + x2

Solving
0 = 22 + 10x + x2

Solving for variable 'x'.

Combine like terms: 0 + -22 = -22
-22 + -10x + -1x2 = 22 + 10x + x2 + -22 + -10x + -1x2

Reorder the terms:
-22 + -10x + -1x2 = 22 + -22 + 10x + -10x + x2 + -1x2

Combine like terms: 22 + -22 = 0
-22 + -10x + -1x2 = 0 + 10x + -10x + x2 + -1x2
-22 + -10x + -1x2 = 10x + -10x + x2 + -1x2

Combine like terms: 10x + -10x = 0
-22 + -10x + -1x2 = 0 + x2 + -1x2
-22 + -10x + -1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
-22 + -10x + -1x2 = 0

Factor out the Greatest Common Factor (GCF), '-1'.
-1(22 + 10x + x2) = 0

Ignore the factor -1.

Subproblem 1
Set the factor '(22 + 10x + x2)' equal to zero and attempt to solve:

Simplifying
22 + 10x + x2 = 0

Solving
22 + 10x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-22' to each side of the equation.
22 + 10x + -22 + x2 = 0 + -22

Reorder the terms:
22 + -22 + 10x + x2 = 0 + -22

Combine like terms: 22 + -22 = 0
0 + 10x + x2 = 0 + -22
10x + x2 = 0 + -22

Combine like terms: 0 + -22 = -22
10x + x2 = -22

The x term is 10x.  Take half its coefficient (5).
Square it (25) and add it to both sides.

Add '25' to each side of the equation.
10x + 25 + x2 = -22 + 25

Reorder the terms:
25 + 10x + x2 = -22 + 25

Combine like terms: -22 + 25 = 3
25 + 10x + x2 = 3

Factor a perfect square on the left side:
(x + 5)(x + 5) = 3

Calculate the square root of the right side: 1.732050808

Break this problem into two subproblems by setting 
(x + 5) equal to 1.732050808 and -1.732050808.


Subproblem 1
x + 5 = 1.732050808

Simplifying
x + 5 = 1.732050808

Reorder the terms:
5 + x = 1.732050808

Solving
5 + x = 1.732050808

Solving for variable 'x'.

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

Add '-5' to each side of the equation.
5 + -5 + x = 1.732050808 + -5

Combine like terms: 5 + -5 = 0
0 + x = 1.732050808 + -5
x = 1.732050808 + -5

Combine like terms: 1.732050808 + -5 = -3.267949192
x = -3.267949192

Simplifying
x = -3.267949192


Subproblem 2
x + 5 = -1.732050808

Simplifying
x + 5 = -1.732050808

Reorder the terms:
5 + x = -1.732050808

Solving
5 + x = -1.732050808

Solving for variable 'x'.

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

Add '-5' to each side of the equation.
5 + -5 + x = -1.732050808 + -5

Combine like terms: 5 + -5 = 0
0 + x = -1.732050808 + -5
x = -1.732050808 + -5

Combine like terms: -1.732050808 + -5 = -6.732050808
x = -6.732050808

Simplifying
x = -6.732050808


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-3.267949192, -6.732050808}
Solution
x = {-3.267949192, -6.732050808}

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