0=x^2+7x+1

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Solution for 0=x^2+7x+1 equation: 


Simplifying
0 = x2 + 7x + 1

Reorder the terms:
0 = 1 + 7x + x2

Solving
0 = 1 + 7x + x2

Solving for variable 'x'.

Combine like terms: 0 + -1 = -1
-1 + -7x + -1x2 = 1 + 7x + x2 + -1 + -7x + -1x2

Reorder the terms:
-1 + -7x + -1x2 = 1 + -1 + 7x + -7x + x2 + -1x2

Combine like terms: 1 + -1 = 0
-1 + -7x + -1x2 = 0 + 7x + -7x + x2 + -1x2
-1 + -7x + -1x2 = 7x + -7x + x2 + -1x2

Combine like terms: 7x + -7x = 0
-1 + -7x + -1x2 = 0 + x2 + -1x2
-1 + -7x + -1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
-1 + -7x + -1x2 = 0

Factor out the Greatest Common Factor (GCF), '-1'.
-1(1 + 7x + x2) = 0

Ignore the factor -1.

Subproblem 1
Set the factor '(1 + 7x + x2)' equal to zero and attempt to solve:

Simplifying
1 + 7x + x2 = 0

Solving
1 + 7x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + 7x + -1 + x2 = 0 + -1

Reorder the terms:
1 + -1 + 7x + x2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + 7x + x2 = 0 + -1
7x + x2 = 0 + -1

Combine like terms: 0 + -1 = -1
7x + x2 = -1

The x term is 7x.  Take half its coefficient (3.5).
Square it (12.25) and add it to both sides.

Add '12.25' to each side of the equation.
7x + 12.25 + x2 = -1 + 12.25

Reorder the terms:
12.25 + 7x + x2 = -1 + 12.25

Combine like terms: -1 + 12.25 = 11.25
12.25 + 7x + x2 = 11.25

Factor a perfect square on the left side:
(x + 3.5)(x + 3.5) = 11.25

Calculate the square root of the right side: 3.354101966

Break this problem into two subproblems by setting 
(x + 3.5) equal to 3.354101966 and -3.354101966.


Subproblem 1
x + 3.5 = 3.354101966

Simplifying
x + 3.5 = 3.354101966

Reorder the terms:
3.5 + x = 3.354101966

Solving
3.5 + x = 3.354101966

Solving for variable 'x'.

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

Add '-3.5' to each side of the equation.
3.5 + -3.5 + x = 3.354101966 + -3.5

Combine like terms: 3.5 + -3.5 = 0.0
0.0 + x = 3.354101966 + -3.5
x = 3.354101966 + -3.5

Combine like terms: 3.354101966 + -3.5 = -0.145898034
x = -0.145898034

Simplifying
x = -0.145898034


Subproblem 2
x + 3.5 = -3.354101966

Simplifying
x + 3.5 = -3.354101966

Reorder the terms:
3.5 + x = -3.354101966

Solving
3.5 + x = -3.354101966

Solving for variable 'x'.

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

Add '-3.5' to each side of the equation.
3.5 + -3.5 + x = -3.354101966 + -3.5

Combine like terms: 3.5 + -3.5 = 0.0
0.0 + x = -3.354101966 + -3.5
x = -3.354101966 + -3.5

Combine like terms: -3.354101966 + -3.5 = -6.854101966
x = -6.854101966

Simplifying
x = -6.854101966


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.145898034, -6.854101966}
Solution
x = {-0.145898034, -6.854101966}

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