6x^2+23x+11=0

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Solution for 6x^2+23x+11=0 equation: 


Simplifying
6x2 + 23x + 11 = 0

Reorder the terms:
11 + 23x + 6x2 = 0

Solving
11 + 23x + 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'.
1.833333333 + 3.833333333x + x2 = 0

Move the constant term to the right:

Add '-1.833333333' to each side of the equation.
1.833333333 + 3.833333333x + -1.833333333 + x2 = 0 + -1.833333333

Reorder the terms:
1.833333333 + -1.833333333 + 3.833333333x + x2 = 0 + -1.833333333

Combine like terms: 1.833333333 + -1.833333333 = 0.000000000
0.000000000 + 3.833333333x + x2 = 0 + -1.833333333
3.833333333x + x2 = 0 + -1.833333333

Combine like terms: 0 + -1.833333333 = -1.833333333
3.833333333x + x2 = -1.833333333

The x term is 3.833333333x.  Take half its coefficient (1.916666667).
Square it (3.673611112) and add it to both sides.

Add '3.673611112' to each side of the equation.
3.833333333x + 3.673611112 + x2 = -1.833333333 + 3.673611112

Reorder the terms:
3.673611112 + 3.833333333x + x2 = -1.833333333 + 3.673611112

Combine like terms: -1.833333333 + 3.673611112 = 1.840277779
3.673611112 + 3.833333333x + x2 = 1.840277779

Factor a perfect square on the left side:
(x + 1.916666667)(x + 1.916666667) = 1.840277779

Calculate the square root of the right side: 1.356568383

Break this problem into two subproblems by setting 
(x + 1.916666667) equal to 1.356568383 and -1.356568383.


Subproblem 1
x + 1.916666667 = 1.356568383

Simplifying
x + 1.916666667 = 1.356568383

Reorder the terms:
1.916666667 + x = 1.356568383

Solving
1.916666667 + x = 1.356568383

Solving for variable 'x'.

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

Add '-1.916666667' to each side of the equation.
1.916666667 + -1.916666667 + x = 1.356568383 + -1.916666667

Combine like terms: 1.916666667 + -1.916666667 = 0.000000000
0.000000000 + x = 1.356568383 + -1.916666667
x = 1.356568383 + -1.916666667

Combine like terms: 1.356568383 + -1.916666667 = -0.560098284
x = -0.560098284

Simplifying
x = -0.560098284


Subproblem 2
x + 1.916666667 = -1.356568383

Simplifying
x + 1.916666667 = -1.356568383

Reorder the terms:
1.916666667 + x = -1.356568383

Solving
1.916666667 + x = -1.356568383

Solving for variable 'x'.

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

Add '-1.916666667' to each side of the equation.
1.916666667 + -1.916666667 + x = -1.356568383 + -1.916666667

Combine like terms: 1.916666667 + -1.916666667 = 0.000000000
0.000000000 + x = -1.356568383 + -1.916666667
x = -1.356568383 + -1.916666667

Combine like terms: -1.356568383 + -1.916666667 = -3.27323505
x = -3.27323505

Simplifying
x = -3.27323505


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.560098284, -3.27323505}

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