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

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


Simplifying
-3x2 + 4x + 6 = 0

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

Solving
6 + 4x + -3x2 = 0

Solving for variable 'x'.

Begin completing the square.  Divide all terms by
-3 the coefficient of the squared term: 

Divide each side by '-3'.
-2 + -1.333333333x + x2 = 0

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + -1.333333333x + 2 + x2 = 0 + 2

Reorder the terms:
-2 + 2 + -1.333333333x + x2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + -1.333333333x + x2 = 0 + 2
-1.333333333x + x2 = 0 + 2

Combine like terms: 0 + 2 = 2
-1.333333333x + x2 = 2

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 = 2 + 0.4444444442

Reorder the terms:
0.4444444442 + -1.333333333x + x2 = 2 + 0.4444444442

Combine like terms: 2 + 0.4444444442 = 2.4444444442
0.4444444442 + -1.333333333x + x2 = 2.4444444442

Factor a perfect square on the left side:
(x + -0.6666666665)(x + -0.6666666665) = 2.4444444442

Calculate the square root of the right side: 1.56347192

Break this problem into two subproblems by setting 
(x + -0.6666666665) equal to 1.56347192 and -1.56347192.


Subproblem 1
x + -0.6666666665 = 1.56347192

Simplifying
x + -0.6666666665 = 1.56347192

Reorder the terms:
-0.6666666665 + x = 1.56347192

Solving
-0.6666666665 + x = 1.56347192

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 = 1.56347192 + 0.6666666665

Combine like terms: -0.6666666665 + 0.6666666665 = 0.0000000000
0.0000000000 + x = 1.56347192 + 0.6666666665
x = 1.56347192 + 0.6666666665

Combine like terms: 1.56347192 + 0.6666666665 = 2.2301385865
x = 2.2301385865

Simplifying
x = 2.2301385865


Subproblem 2
x + -0.6666666665 = -1.56347192

Simplifying
x + -0.6666666665 = -1.56347192

Reorder the terms:
-0.6666666665 + x = -1.56347192

Solving
-0.6666666665 + x = -1.56347192

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 = -1.56347192 + 0.6666666665

Combine like terms: -0.6666666665 + 0.6666666665 = 0.0000000000
0.0000000000 + x = -1.56347192 + 0.6666666665
x = -1.56347192 + 0.6666666665

Combine like terms: -1.56347192 + 0.6666666665 = -0.8968052535
x = -0.8968052535

Simplifying
x = -0.8968052535


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.2301385865, -0.8968052535}

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