x^2+10x+22=31

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Solution for x^2+10x+22=31 equation: 


Simplifying
x2 + 10x + 22 = 31

Reorder the terms:
22 + 10x + x2 = 31

Solving
22 + 10x + x2 = 31

Solving for variable 'x'.

Reorder the terms:
22 + -31 + 10x + x2 = 31 + -31

Combine like terms: 22 + -31 = -9
-9 + 10x + x2 = 31 + -31

Combine like terms: 31 + -31 = 0
-9 + 10x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '9' to each side of the equation.
-9 + 10x + 9 + x2 = 0 + 9

Reorder the terms:
-9 + 9 + 10x + x2 = 0 + 9

Combine like terms: -9 + 9 = 0
0 + 10x + x2 = 0 + 9
10x + x2 = 0 + 9

Combine like terms: 0 + 9 = 9
10x + x2 = 9

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 = 9 + 25

Reorder the terms:
25 + 10x + x2 = 9 + 25

Combine like terms: 9 + 25 = 34
25 + 10x + x2 = 34

Factor a perfect square on the left side:
(x + 5)(x + 5) = 34

Calculate the square root of the right side: 5.830951895

Break this problem into two subproblems by setting 
(x + 5) equal to 5.830951895 and -5.830951895.


Subproblem 1
x + 5 = 5.830951895

Simplifying
x + 5 = 5.830951895

Reorder the terms:
5 + x = 5.830951895

Solving
5 + x = 5.830951895

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 = 5.830951895 + -5

Combine like terms: 5 + -5 = 0
0 + x = 5.830951895 + -5
x = 5.830951895 + -5

Combine like terms: 5.830951895 + -5 = 0.830951895
x = 0.830951895

Simplifying
x = 0.830951895


Subproblem 2
x + 5 = -5.830951895

Simplifying
x + 5 = -5.830951895

Reorder the terms:
5 + x = -5.830951895

Solving
5 + x = -5.830951895

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 = -5.830951895 + -5

Combine like terms: 5 + -5 = 0
0 + x = -5.830951895 + -5
x = -5.830951895 + -5

Combine like terms: -5.830951895 + -5 = -10.830951895
x = -10.830951895

Simplifying
x = -10.830951895


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.830951895, -10.830951895}

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