6x^2=8-5x

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Solution for 6x^2=8-5x equation: 


Simplifying
6x2 = 8 + -5x

Solving
6x2 = 8 + -5x

Solving for variable 'x'.

Reorder the terms:
-8 + 5x + 6x2 = 8 + -5x + -8 + 5x

Reorder the terms:
-8 + 5x + 6x2 = 8 + -8 + -5x + 5x

Combine like terms: 8 + -8 = 0
-8 + 5x + 6x2 = 0 + -5x + 5x
-8 + 5x + 6x2 = -5x + 5x

Combine like terms: -5x + 5x = 0
-8 + 5x + 6x2 = 0

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

Divide each side by '6'.
-1.333333333 + 0.8333333333x + x2 = 0

Move the constant term to the right:

Add '1.333333333' to each side of the equation.
-1.333333333 + 0.8333333333x + 1.333333333 + x2 = 0 + 1.333333333

Reorder the terms:
-1.333333333 + 1.333333333 + 0.8333333333x + x2 = 0 + 1.333333333

Combine like terms: -1.333333333 + 1.333333333 = 0.000000000
0.000000000 + 0.8333333333x + x2 = 0 + 1.333333333
0.8333333333x + x2 = 0 + 1.333333333

Combine like terms: 0 + 1.333333333 = 1.333333333
0.8333333333x + x2 = 1.333333333

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 = 1.333333333 + 0.1736111111

Reorder the terms:
0.1736111111 + 0.8333333333x + x2 = 1.333333333 + 0.1736111111

Combine like terms: 1.333333333 + 0.1736111111 = 1.5069444441
0.1736111111 + 0.8333333333x + x2 = 1.5069444441

Factor a perfect square on the left side:
(x + 0.4166666667)(x + 0.4166666667) = 1.5069444441

Calculate the square root of the right side: 1.227576655

Break this problem into two subproblems by setting 
(x + 0.4166666667) equal to 1.227576655 and -1.227576655.


Subproblem 1
x + 0.4166666667 = 1.227576655

Simplifying
x + 0.4166666667 = 1.227576655

Reorder the terms:
0.4166666667 + x = 1.227576655

Solving
0.4166666667 + x = 1.227576655

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 = 1.227576655 + -0.4166666667

Combine like terms: 0.4166666667 + -0.4166666667 = 0.0000000000
0.0000000000 + x = 1.227576655 + -0.4166666667
x = 1.227576655 + -0.4166666667

Combine like terms: 1.227576655 + -0.4166666667 = 0.8109099883
x = 0.8109099883

Simplifying
x = 0.8109099883


Subproblem 2
x + 0.4166666667 = -1.227576655

Simplifying
x + 0.4166666667 = -1.227576655

Reorder the terms:
0.4166666667 + x = -1.227576655

Solving
0.4166666667 + x = -1.227576655

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 = -1.227576655 + -0.4166666667

Combine like terms: 0.4166666667 + -0.4166666667 = 0.0000000000
0.0000000000 + x = -1.227576655 + -0.4166666667
x = -1.227576655 + -0.4166666667

Combine like terms: -1.227576655 + -0.4166666667 = -1.6442433217
x = -1.6442433217

Simplifying
x = -1.6442433217


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.8109099883, -1.6442433217}

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