6x^2=8x-1

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Solution for 6x^2=8x-1 equation:


Simplifying
6x2 = 8x + -1

Reorder the terms:
6x2 = -1 + 8x

Solving
6x2 = -1 + 8x

Solving for variable 'x'.

Reorder the terms:
1 + -8x + 6x2 = -1 + 8x + 1 + -8x

Reorder the terms:
1 + -8x + 6x2 = -1 + 1 + 8x + -8x

Combine like terms: -1 + 1 = 0
1 + -8x + 6x2 = 0 + 8x + -8x
1 + -8x + 6x2 = 8x + -8x

Combine like terms: 8x + -8x = 0
1 + -8x + 6x2 = 0

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

Divide each side by '6'.
0.1666666667 + -1.333333333x + x2 = 0

Move the constant term to the right:

Add '-0.1666666667' to each side of the equation.
0.1666666667 + -1.333333333x + -0.1666666667 + x2 = 0 + -0.1666666667

Reorder the terms:
0.1666666667 + -0.1666666667 + -1.333333333x + x2 = 0 + -0.1666666667

Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000
0.0000000000 + -1.333333333x + x2 = 0 + -0.1666666667
-1.333333333x + x2 = 0 + -0.1666666667

Combine like terms: 0 + -0.1666666667 = -0.1666666667
-1.333333333x + x2 = -0.1666666667

The x term is -1.333333333x.  Take half its coefficient (-0.6666666665).
Square it (0.4444444442) and add it to both sides.

Add '0.4444444442' to each side of the equation.
-1.333333333x + 0.4444444442 + x2 = -0.1666666667 + 0.4444444442

Reorder the terms:
0.4444444442 + -1.333333333x + x2 = -0.1666666667 + 0.4444444442

Combine like terms: -0.1666666667 + 0.4444444442 = 0.2777777775
0.4444444442 + -1.333333333x + x2 = 0.2777777775

Factor a perfect square on the left side:
(x + -0.6666666665)(x + -0.6666666665) = 0.2777777775

Calculate the square root of the right side: 0.527046276

Break this problem into two subproblems by setting 
(x + -0.6666666665) equal to 0.527046276 and -0.527046276.

Subproblem 1

x + -0.6666666665 = 0.527046276 Simplifying x + -0.6666666665 = 0.527046276 Reorder the terms: -0.6666666665 + x = 0.527046276 Solving -0.6666666665 + x = 0.527046276 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.6666666665' to each side of the equation. -0.6666666665 + 0.6666666665 + x = 0.527046276 + 0.6666666665 Combine like terms: -0.6666666665 + 0.6666666665 = 0.0000000000 0.0000000000 + x = 0.527046276 + 0.6666666665 x = 0.527046276 + 0.6666666665 Combine like terms: 0.527046276 + 0.6666666665 = 1.1937129425 x = 1.1937129425 Simplifying x = 1.1937129425

Subproblem 2

x + -0.6666666665 = -0.527046276 Simplifying x + -0.6666666665 = -0.527046276 Reorder the terms: -0.6666666665 + x = -0.527046276 Solving -0.6666666665 + x = -0.527046276 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.6666666665' to each side of the equation. -0.6666666665 + 0.6666666665 + x = -0.527046276 + 0.6666666665 Combine like terms: -0.6666666665 + 0.6666666665 = 0.0000000000 0.0000000000 + x = -0.527046276 + 0.6666666665 x = -0.527046276 + 0.6666666665 Combine like terms: -0.527046276 + 0.6666666665 = 0.1396203905 x = 0.1396203905 Simplifying x = 0.1396203905

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.1937129425, 0.1396203905}

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