x+1=6x^2+18x+5

Solution for x+1=6x^2+18x+5 equation:

Simplifying
x + 1 = 6x2 + 18x + 5

Reorder the terms:
1 + x = 6x2 + 18x + 5

Reorder the terms:
1 + x = 5 + 18x + 6x2

Solving
1 + x = 5 + 18x + 6x2

Solving for variable 'x'.

Reorder the terms:
1 + -5 + x + -18x + -6x2 = 5 + 18x + 6x2 + -5 + -18x + -6x2

Combine like terms: 1 + -5 = -4
-4 + x + -18x + -6x2 = 5 + 18x + 6x2 + -5 + -18x + -6x2

Combine like terms: x + -18x = -17x
-4 + -17x + -6x2 = 5 + 18x + 6x2 + -5 + -18x + -6x2

Reorder the terms:
-4 + -17x + -6x2 = 5 + -5 + 18x + -18x + 6x2 + -6x2

Combine like terms: 5 + -5 = 0
-4 + -17x + -6x2 = 0 + 18x + -18x + 6x2 + -6x2
-4 + -17x + -6x2 = 18x + -18x + 6x2 + -6x2

Combine like terms: 18x + -18x = 0
-4 + -17x + -6x2 = 0 + 6x2 + -6x2
-4 + -17x + -6x2 = 6x2 + -6x2

Combine like terms: 6x2 + -6x2 = 0
-4 + -17x + -6x2 = 0

Factor out the Greatest Common Factor (GCF), '-1'.
-1(4 + 17x + 6x2) = 0

Ignore the factor -1.
Subproblem 1
Set the factor '(4 + 17x + 6x2)' equal to zero and attempt to solve:

Simplifying
4 + 17x + 6x2 = 0

Solving
4 + 17x + 6x2 = 0

Begin completing the square.  Divide all terms by
6 the coefficient of the squared term: 

Divide each side by '6'.
0.6666666667 + 2.833333333x + x2 = 0

Move the constant term to the right:

Add '-0.6666666667' to each side of the equation.
0.6666666667 + 2.833333333x + -0.6666666667 + x2 = 0 + -0.6666666667

Reorder the terms:
0.6666666667 + -0.6666666667 + 2.833333333x + x2 = 0 + -0.6666666667

Combine like terms: 0.6666666667 + -0.6666666667 = 0.0000000000
0.0000000000 + 2.833333333x + x2 = 0 + -0.6666666667
2.833333333x + x2 = 0 + -0.6666666667

Combine like terms: 0 + -0.6666666667 = -0.6666666667
2.833333333x + x2 = -0.6666666667

The x term is 2.833333333x.  Take half its coefficient (1.416666667).
Square it (2.006944445) and add it to both sides.

Add '2.006944445' to each side of the equation.
2.833333333x + 2.006944445 + x2 = -0.6666666667 + 2.006944445

Reorder the terms:
2.006944445 + 2.833333333x + x2 = -0.6666666667 + 2.006944445

Combine like terms: -0.6666666667 + 2.006944445 = 1.3402777783
2.006944445 + 2.833333333x + x2 = 1.3402777783

Factor a perfect square on the left side:
(x + 1.416666667)(x + 1.416666667) = 1.3402777783

Calculate the square root of the right side: 1.157703666

Break this problem into two subproblems by setting 
(x + 1.416666667) equal to 1.157703666 and -1.157703666.

Subproblem 1
x + 1.416666667 = 1.157703666

Simplifying
x + 1.416666667 = 1.157703666

Reorder the terms:
1.416666667 + x = 1.157703666

Solving
1.416666667 + x = 1.157703666

Solving for variable 'x'.

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

Add '-1.416666667' to each side of the equation.
1.416666667 + -1.416666667 + x = 1.157703666 + -1.416666667

Combine like terms: 1.416666667 + -1.416666667 = 0.000000000
0.000000000 + x = 1.157703666 + -1.416666667
x = 1.157703666 + -1.416666667

Combine like terms: 1.157703666 + -1.416666667 = -0.258963001
x = -0.258963001

Simplifying
x = -0.258963001

Subproblem 2
x + 1.416666667 = -1.157703666

Simplifying
x + 1.416666667 = -1.157703666

Reorder the terms:
1.416666667 + x = -1.157703666

Solving
1.416666667 + x = -1.157703666

Solving for variable 'x'.

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

Add '-1.416666667' to each side of the equation.
1.416666667 + -1.416666667 + x = -1.157703666 + -1.416666667

Combine like terms: 1.416666667 + -1.416666667 = 0.000000000
0.000000000 + x = -1.157703666 + -1.416666667
x = -1.157703666 + -1.416666667

Combine like terms: -1.157703666 + -1.416666667 = -2.574370333
x = -2.574370333

Simplifying
x = -2.574370333

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.258963001, -2.574370333}
Solution
x = {-0.258963001, -2.574370333}