-6x^2-4x+3=0

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Solution for -6x^2-4x+3=0 equation: 


Simplifying
-6x2 + -4x + 3 = 0

Reorder the terms:
3 + -4x + -6x2 = 0

Solving
3 + -4x + -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'.
-0.5 + 0.6666666667x + x2 = 0

Move the constant term to the right:

Add '0.5' to each side of the equation.
-0.5 + 0.6666666667x + 0.5 + x2 = 0 + 0.5

Reorder the terms:
-0.5 + 0.5 + 0.6666666667x + x2 = 0 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + 0.6666666667x + x2 = 0 + 0.5
0.6666666667x + x2 = 0 + 0.5

Combine like terms: 0 + 0.5 = 0.5
0.6666666667x + x2 = 0.5

The x term is 0.6666666667x.  Take half its coefficient (0.3333333334).
Square it (0.1111111112) and add it to both sides.

Add '0.1111111112' to each side of the equation.
0.6666666667x + 0.1111111112 + x2 = 0.5 + 0.1111111112

Reorder the terms:
0.1111111112 + 0.6666666667x + x2 = 0.5 + 0.1111111112

Combine like terms: 0.5 + 0.1111111112 = 0.6111111112
0.1111111112 + 0.6666666667x + x2 = 0.6111111112

Factor a perfect square on the left side:
(x + 0.3333333334)(x + 0.3333333334) = 0.6111111112

Calculate the square root of the right side: 0.78173596

Break this problem into two subproblems by setting 
(x + 0.3333333334) equal to 0.78173596 and -0.78173596.


Subproblem 1
x + 0.3333333334 = 0.78173596

Simplifying
x + 0.3333333334 = 0.78173596

Reorder the terms:
0.3333333334 + x = 0.78173596

Solving
0.3333333334 + x = 0.78173596

Solving for variable 'x'.

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

Add '-0.3333333334' to each side of the equation.
0.3333333334 + -0.3333333334 + x = 0.78173596 + -0.3333333334

Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000
0.0000000000 + x = 0.78173596 + -0.3333333334
x = 0.78173596 + -0.3333333334

Combine like terms: 0.78173596 + -0.3333333334 = 0.4484026266
x = 0.4484026266

Simplifying
x = 0.4484026266


Subproblem 2
x + 0.3333333334 = -0.78173596

Simplifying
x + 0.3333333334 = -0.78173596

Reorder the terms:
0.3333333334 + x = -0.78173596

Solving
0.3333333334 + x = -0.78173596

Solving for variable 'x'.

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

Add '-0.3333333334' to each side of the equation.
0.3333333334 + -0.3333333334 + x = -0.78173596 + -0.3333333334

Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000
0.0000000000 + x = -0.78173596 + -0.3333333334
x = -0.78173596 + -0.3333333334

Combine like terms: -0.78173596 + -0.3333333334 = -1.1150692934
x = -1.1150692934

Simplifying
x = -1.1150692934


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.4484026266, -1.1150692934}

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