6x^2+5x-27-6=0

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Solution for 6x^2+5x-27-6=0 equation: 


Simplifying
6x2 + 5x + -27 + -6 = 0

Reorder the terms:
-27 + -6 + 5x + 6x2 = 0

Combine like terms: -27 + -6 = -33
-33 + 5x + 6x2 = 0

Solving
-33 + 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'.
-5.5 + 0.8333333333x + x2 = 0

Move the constant term to the right:

Add '5.5' to each side of the equation.
-5.5 + 0.8333333333x + 5.5 + x2 = 0 + 5.5

Reorder the terms:
-5.5 + 5.5 + 0.8333333333x + x2 = 0 + 5.5

Combine like terms: -5.5 + 5.5 = 0.0
0.0 + 0.8333333333x + x2 = 0 + 5.5
0.8333333333x + x2 = 0 + 5.5

Combine like terms: 0 + 5.5 = 5.5
0.8333333333x + x2 = 5.5

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 = 5.5 + 0.1736111111

Reorder the terms:
0.1736111111 + 0.8333333333x + x2 = 5.5 + 0.1736111111

Combine like terms: 5.5 + 0.1736111111 = 5.6736111111
0.1736111111 + 0.8333333333x + x2 = 5.6736111111

Factor a perfect square on the left side:
(x + 0.4166666667)(x + 0.4166666667) = 5.6736111111

Calculate the square root of the right side: 2.381934321

Break this problem into two subproblems by setting 
(x + 0.4166666667) equal to 2.381934321 and -2.381934321.


Subproblem 1
x + 0.4166666667 = 2.381934321

Simplifying
x + 0.4166666667 = 2.381934321

Reorder the terms:
0.4166666667 + x = 2.381934321

Solving
0.4166666667 + x = 2.381934321

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 = 2.381934321 + -0.4166666667

Combine like terms: 0.4166666667 + -0.4166666667 = 0.0000000000
0.0000000000 + x = 2.381934321 + -0.4166666667
x = 2.381934321 + -0.4166666667

Combine like terms: 2.381934321 + -0.4166666667 = 1.9652676543
x = 1.9652676543

Simplifying
x = 1.9652676543


Subproblem 2
x + 0.4166666667 = -2.381934321

Simplifying
x + 0.4166666667 = -2.381934321

Reorder the terms:
0.4166666667 + x = -2.381934321

Solving
0.4166666667 + x = -2.381934321

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 = -2.381934321 + -0.4166666667

Combine like terms: 0.4166666667 + -0.4166666667 = 0.0000000000
0.0000000000 + x = -2.381934321 + -0.4166666667
x = -2.381934321 + -0.4166666667

Combine like terms: -2.381934321 + -0.4166666667 = -2.7986009877
x = -2.7986009877

Simplifying
x = -2.7986009877


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.9652676543, -2.7986009877}

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