0=x^2+10x+17

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Solution for 0=x^2+10x+17 equation: 


Simplifying
0 = x2 + 10x + 17

Reorder the terms:
0 = 17 + 10x + x2

Solving
0 = 17 + 10x + x2

Solving for variable 'x'.

Combine like terms: 0 + -17 = -17
-17 + -10x + -1x2 = 17 + 10x + x2 + -17 + -10x + -1x2

Reorder the terms:
-17 + -10x + -1x2 = 17 + -17 + 10x + -10x + x2 + -1x2

Combine like terms: 17 + -17 = 0
-17 + -10x + -1x2 = 0 + 10x + -10x + x2 + -1x2
-17 + -10x + -1x2 = 10x + -10x + x2 + -1x2

Combine like terms: 10x + -10x = 0
-17 + -10x + -1x2 = 0 + x2 + -1x2
-17 + -10x + -1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
-17 + -10x + -1x2 = 0

Factor out the Greatest Common Factor (GCF), '-1'.
-1(17 + 10x + x2) = 0

Ignore the factor -1.

Subproblem 1
Set the factor '(17 + 10x + x2)' equal to zero and attempt to solve:

Simplifying
17 + 10x + x2 = 0

Solving
17 + 10x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-17' to each side of the equation.
17 + 10x + -17 + x2 = 0 + -17

Reorder the terms:
17 + -17 + 10x + x2 = 0 + -17

Combine like terms: 17 + -17 = 0
0 + 10x + x2 = 0 + -17
10x + x2 = 0 + -17

Combine like terms: 0 + -17 = -17
10x + x2 = -17

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 = -17 + 25

Reorder the terms:
25 + 10x + x2 = -17 + 25

Combine like terms: -17 + 25 = 8
25 + 10x + x2 = 8

Factor a perfect square on the left side:
(x + 5)(x + 5) = 8

Calculate the square root of the right side: 2.828427125

Break this problem into two subproblems by setting 
(x + 5) equal to 2.828427125 and -2.828427125.


Subproblem 1
x + 5 = 2.828427125

Simplifying
x + 5 = 2.828427125

Reorder the terms:
5 + x = 2.828427125

Solving
5 + x = 2.828427125

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 = 2.828427125 + -5

Combine like terms: 5 + -5 = 0
0 + x = 2.828427125 + -5
x = 2.828427125 + -5

Combine like terms: 2.828427125 + -5 = -2.171572875
x = -2.171572875

Simplifying
x = -2.171572875


Subproblem 2
x + 5 = -2.828427125

Simplifying
x + 5 = -2.828427125

Reorder the terms:
5 + x = -2.828427125

Solving
5 + x = -2.828427125

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 = -2.828427125 + -5

Combine like terms: 5 + -5 = 0
0 + x = -2.828427125 + -5
x = -2.828427125 + -5

Combine like terms: -2.828427125 + -5 = -7.828427125
x = -7.828427125

Simplifying
x = -7.828427125


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-2.171572875, -7.828427125}
Solution
x = {-2.171572875, -7.828427125}

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