(6x+5)(6x+5)-125=0

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Solution for (6x+5)(6x+5)-125=0 equation: 


Simplifying
(6x + 5)(6x + 5) + -125 = 0

Reorder the terms:
(5 + 6x)(6x + 5) + -125 = 0

Reorder the terms:
(5 + 6x)(5 + 6x) + -125 = 0

Multiply (5 + 6x) * (5 + 6x)
(5(5 + 6x) + 6x * (5 + 6x)) + -125 = 0
((5 * 5 + 6x * 5) + 6x * (5 + 6x)) + -125 = 0
((25 + 30x) + 6x * (5 + 6x)) + -125 = 0
(25 + 30x + (5 * 6x + 6x * 6x)) + -125 = 0
(25 + 30x + (30x + 36x2)) + -125 = 0

Combine like terms: 30x + 30x = 60x
(25 + 60x + 36x2) + -125 = 0

Reorder the terms:
25 + -125 + 60x + 36x2 = 0

Combine like terms: 25 + -125 = -100
-100 + 60x + 36x2 = 0

Solving
-100 + 60x + 36x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '4'.
4(-25 + 15x + 9x2) = 0

Ignore the factor 4.

Subproblem 1
Set the factor '(-25 + 15x + 9x2)' equal to zero and attempt to solve:

Simplifying
-25 + 15x + 9x2 = 0

Solving
-25 + 15x + 9x2 = 0

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

Divide each side by '9'.
-2.777777778 + 1.666666667x + x2 = 0

Move the constant term to the right:

Add '2.777777778' to each side of the equation.
-2.777777778 + 1.666666667x + 2.777777778 + x2 = 0 + 2.777777778

Reorder the terms:
-2.777777778 + 2.777777778 + 1.666666667x + x2 = 0 + 2.777777778

Combine like terms: -2.777777778 + 2.777777778 = 0.000000000
0.000000000 + 1.666666667x + x2 = 0 + 2.777777778
1.666666667x + x2 = 0 + 2.777777778

Combine like terms: 0 + 2.777777778 = 2.777777778
1.666666667x + x2 = 2.777777778

The x term is 1.666666667x.  Take half its coefficient (0.8333333335).
Square it (0.6944444447) and add it to both sides.

Add '0.6944444447' to each side of the equation.
1.666666667x + 0.6944444447 + x2 = 2.777777778 + 0.6944444447

Reorder the terms:
0.6944444447 + 1.666666667x + x2 = 2.777777778 + 0.6944444447

Combine like terms: 2.777777778 + 0.6944444447 = 3.4722222227
0.6944444447 + 1.666666667x + x2 = 3.4722222227

Factor a perfect square on the left side:
(x + 0.8333333335)(x + 0.8333333335) = 3.4722222227

Calculate the square root of the right side: 1.863389981

Break this problem into two subproblems by setting 
(x + 0.8333333335) equal to 1.863389981 and -1.863389981.


Subproblem 1
x + 0.8333333335 = 1.863389981

Simplifying
x + 0.8333333335 = 1.863389981

Reorder the terms:
0.8333333335 + x = 1.863389981

Solving
0.8333333335 + x = 1.863389981

Solving for variable 'x'.

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

Add '-0.8333333335' to each side of the equation.
0.8333333335 + -0.8333333335 + x = 1.863389981 + -0.8333333335

Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000
0.0000000000 + x = 1.863389981 + -0.8333333335
x = 1.863389981 + -0.8333333335

Combine like terms: 1.863389981 + -0.8333333335 = 1.0300566475
x = 1.0300566475

Simplifying
x = 1.0300566475


Subproblem 2
x + 0.8333333335 = -1.863389981

Simplifying
x + 0.8333333335 = -1.863389981

Reorder the terms:
0.8333333335 + x = -1.863389981

Solving
0.8333333335 + x = -1.863389981

Solving for variable 'x'.

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

Add '-0.8333333335' to each side of the equation.
0.8333333335 + -0.8333333335 + x = -1.863389981 + -0.8333333335

Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000
0.0000000000 + x = -1.863389981 + -0.8333333335
x = -1.863389981 + -0.8333333335

Combine like terms: -1.863389981 + -0.8333333335 = -2.6967233145
x = -2.6967233145

Simplifying
x = -2.6967233145


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.0300566475, -2.6967233145}
Solution
x = {1.0300566475, -2.6967233145}

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