2x+13=x^2+14x+14

Solution for 2x+13=x^2+14x+14 equation:

Simplifying
2x + 13 = x2 + 14x + 14

Reorder the terms:
13 + 2x = x2 + 14x + 14

Reorder the terms:
13 + 2x = 14 + 14x + x2

Solving
13 + 2x = 14 + 14x + x2

Solving for variable 'x'.

Reorder the terms:
13 + -14 + 2x + -14x + -1x2 = 14 + 14x + x2 + -14 + -14x + -1x2

Combine like terms: 13 + -14 = -1
-1 + 2x + -14x + -1x2 = 14 + 14x + x2 + -14 + -14x + -1x2

Combine like terms: 2x + -14x = -12x
-1 + -12x + -1x2 = 14 + 14x + x2 + -14 + -14x + -1x2

Reorder the terms:
-1 + -12x + -1x2 = 14 + -14 + 14x + -14x + x2 + -1x2

Combine like terms: 14 + -14 = 0
-1 + -12x + -1x2 = 0 + 14x + -14x + x2 + -1x2
-1 + -12x + -1x2 = 14x + -14x + x2 + -1x2

Combine like terms: 14x + -14x = 0
-1 + -12x + -1x2 = 0 + x2 + -1x2
-1 + -12x + -1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
-1 + -12x + -1x2 = 0

Factor out the Greatest Common Factor (GCF), '-1'.
-1(1 + 12x + x2) = 0

Ignore the factor -1.
Subproblem 1
Set the factor '(1 + 12x + x2)' equal to zero and attempt to solve:

Simplifying
1 + 12x + x2 = 0

Solving
1 + 12x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + 12x + -1 + x2 = 0 + -1

Reorder the terms:
1 + -1 + 12x + x2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + 12x + x2 = 0 + -1
12x + x2 = 0 + -1

Combine like terms: 0 + -1 = -1
12x + x2 = -1

The x term is 12x.  Take half its coefficient (6).
Square it (36) and add it to both sides.

Add '36' to each side of the equation.
12x + 36 + x2 = -1 + 36

Reorder the terms:
36 + 12x + x2 = -1 + 36

Combine like terms: -1 + 36 = 35
36 + 12x + x2 = 35

Factor a perfect square on the left side:
(x + 6)(x + 6) = 35

Calculate the square root of the right side: 5.916079783

Break this problem into two subproblems by setting 
(x + 6) equal to 5.916079783 and -5.916079783.

Subproblem 1
x + 6 = 5.916079783

Simplifying
x + 6 = 5.916079783

Reorder the terms:
6 + x = 5.916079783

Solving
6 + x = 5.916079783

Solving for variable 'x'.

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

Add '-6' to each side of the equation.
6 + -6 + x = 5.916079783 + -6

Combine like terms: 6 + -6 = 0
0 + x = 5.916079783 + -6
x = 5.916079783 + -6

Combine like terms: 5.916079783 + -6 = -0.083920217
x = -0.083920217

Simplifying
x = -0.083920217

Subproblem 2
x + 6 = -5.916079783

Simplifying
x + 6 = -5.916079783

Reorder the terms:
6 + x = -5.916079783

Solving
6 + x = -5.916079783

Solving for variable 'x'.

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

Add '-6' to each side of the equation.
6 + -6 + x = -5.916079783 + -6

Combine like terms: 6 + -6 = 0
0 + x = -5.916079783 + -6
x = -5.916079783 + -6

Combine like terms: -5.916079783 + -6 = -11.916079783
x = -11.916079783

Simplifying
x = -11.916079783

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.083920217, -11.916079783}
Solution
x = {-0.083920217, -11.916079783}