6x^2+5x-229=0

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Solution for 6x^2+5x-229=0 equation: 


Simplifying
6x2 + 5x + -229 = 0

Reorder the terms:
-229 + 5x + 6x2 = 0

Solving
-229 + 5x + 6x2 = 0

Solving for variable 'x'.

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

Divide each side by '6'.
-38.16666667 + 0.8333333333x + x2 = 0

Move the constant term to the right:

Add '38.16666667' to each side of the equation.
-38.16666667 + 0.8333333333x + 38.16666667 + x2 = 0 + 38.16666667

Reorder the terms:
-38.16666667 + 38.16666667 + 0.8333333333x + x2 = 0 + 38.16666667

Combine like terms: -38.16666667 + 38.16666667 = 0.00000000
0.00000000 + 0.8333333333x + x2 = 0 + 38.16666667
0.8333333333x + x2 = 0 + 38.16666667

Combine like terms: 0 + 38.16666667 = 38.16666667
0.8333333333x + x2 = 38.16666667

The x term is 0.8333333333x.  Take half its coefficient (0.4166666667).
Square it (0.1736111111) and add it to both sides.

Add '0.1736111111' to each side of the equation.
0.8333333333x + 0.1736111111 + x2 = 38.16666667 + 0.1736111111

Reorder the terms:
0.1736111111 + 0.8333333333x + x2 = 38.16666667 + 0.1736111111

Combine like terms: 38.16666667 + 0.1736111111 = 38.3402777811
0.1736111111 + 0.8333333333x + x2 = 38.3402777811

Factor a perfect square on the left side:
(x + 0.4166666667)(x + 0.4166666667) = 38.3402777811

Calculate the square root of the right side: 6.191952663

Break this problem into two subproblems by setting 
(x + 0.4166666667) equal to 6.191952663 and -6.191952663.


Subproblem 1
x + 0.4166666667 = 6.191952663

Simplifying
x + 0.4166666667 = 6.191952663

Reorder the terms:
0.4166666667 + x = 6.191952663

Solving
0.4166666667 + x = 6.191952663

Solving for variable 'x'.

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

Add '-0.4166666667' to each side of the equation.
0.4166666667 + -0.4166666667 + x = 6.191952663 + -0.4166666667

Combine like terms: 0.4166666667 + -0.4166666667 = 0.0000000000
0.0000000000 + x = 6.191952663 + -0.4166666667
x = 6.191952663 + -0.4166666667

Combine like terms: 6.191952663 + -0.4166666667 = 5.7752859963
x = 5.7752859963

Simplifying
x = 5.7752859963


Subproblem 2
x + 0.4166666667 = -6.191952663

Simplifying
x + 0.4166666667 = -6.191952663

Reorder the terms:
0.4166666667 + x = -6.191952663

Solving
0.4166666667 + x = -6.191952663

Solving for variable 'x'.

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

Add '-0.4166666667' to each side of the equation.
0.4166666667 + -0.4166666667 + x = -6.191952663 + -0.4166666667

Combine like terms: 0.4166666667 + -0.4166666667 = 0.0000000000
0.0000000000 + x = -6.191952663 + -0.4166666667
x = -6.191952663 + -0.4166666667

Combine like terms: -6.191952663 + -0.4166666667 = -6.6086193297
x = -6.6086193297

Simplifying
x = -6.6086193297


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {5.7752859963, -6.6086193297}

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