12x^2-6x-3=0

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Solution for 12x^2-6x-3=0 equation: 


Simplifying
12x2 + -6x + -3 = 0

Reorder the terms:
-3 + -6x + 12x2 = 0

Solving
-3 + -6x + 12x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '3'.
3(-1 + -2x + 4x2) = 0

Ignore the factor 3.

Subproblem 1
Set the factor '(-1 + -2x + 4x2)' equal to zero and attempt to solve:

Simplifying
-1 + -2x + 4x2 = 0

Solving
-1 + -2x + 4x2 = 0

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

Divide each side by '4'.
-0.25 + -0.5x + x2 = 0

Move the constant term to the right:

Add '0.25' to each side of the equation.
-0.25 + -0.5x + 0.25 + x2 = 0 + 0.25

Reorder the terms:
-0.25 + 0.25 + -0.5x + x2 = 0 + 0.25

Combine like terms: -0.25 + 0.25 = 0.00
0.00 + -0.5x + x2 = 0 + 0.25
-0.5x + x2 = 0 + 0.25

Combine like terms: 0 + 0.25 = 0.25
-0.5x + x2 = 0.25

The x term is -0.5x.  Take half its coefficient (-0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
-0.5x + 0.0625 + x2 = 0.25 + 0.0625

Reorder the terms:
0.0625 + -0.5x + x2 = 0.25 + 0.0625

Combine like terms: 0.25 + 0.0625 = 0.3125
0.0625 + -0.5x + x2 = 0.3125

Factor a perfect square on the left side:
(x + -0.25)(x + -0.25) = 0.3125

Calculate the square root of the right side: 0.559016994

Break this problem into two subproblems by setting 
(x + -0.25) equal to 0.559016994 and -0.559016994.


Subproblem 1
x + -0.25 = 0.559016994

Simplifying
x + -0.25 = 0.559016994

Reorder the terms:
-0.25 + x = 0.559016994

Solving
-0.25 + x = 0.559016994

Solving for variable 'x'.

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

Add '0.25' to each side of the equation.
-0.25 + 0.25 + x = 0.559016994 + 0.25

Combine like terms: -0.25 + 0.25 = 0.00
0.00 + x = 0.559016994 + 0.25
x = 0.559016994 + 0.25

Combine like terms: 0.559016994 + 0.25 = 0.809016994
x = 0.809016994

Simplifying
x = 0.809016994


Subproblem 2
x + -0.25 = -0.559016994

Simplifying
x + -0.25 = -0.559016994

Reorder the terms:
-0.25 + x = -0.559016994

Solving
-0.25 + x = -0.559016994

Solving for variable 'x'.

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

Add '0.25' to each side of the equation.
-0.25 + 0.25 + x = -0.559016994 + 0.25

Combine like terms: -0.25 + 0.25 = 0.00
0.00 + x = -0.559016994 + 0.25
x = -0.559016994 + 0.25

Combine like terms: -0.559016994 + 0.25 = -0.309016994
x = -0.309016994

Simplifying
x = -0.309016994


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.809016994, -0.309016994}
Solution
x = {0.809016994, -0.309016994}

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