12x^2-52x+3=0

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Solution for 12x^2-52x+3=0 equation: 


Simplifying
12x2 + -52x + 3 = 0

Reorder the terms:
3 + -52x + 12x2 = 0

Solving
3 + -52x + 12x2 = 0

Solving for variable 'x'.

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

Divide each side by '12'.
0.25 + -4.333333333x + x2 = 0

Move the constant term to the right:

Add '-0.25' to each side of the equation.
0.25 + -4.333333333x + -0.25 + x2 = 0 + -0.25

Reorder the terms:
0.25 + -0.25 + -4.333333333x + x2 = 0 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + -4.333333333x + x2 = 0 + -0.25
-4.333333333x + x2 = 0 + -0.25

Combine like terms: 0 + -0.25 = -0.25
-4.333333333x + x2 = -0.25

The x term is -4.333333333x.  Take half its coefficient (-2.166666667).
Square it (4.694444446) and add it to both sides.

Add '4.694444446' to each side of the equation.
-4.333333333x + 4.694444446 + x2 = -0.25 + 4.694444446

Reorder the terms:
4.694444446 + -4.333333333x + x2 = -0.25 + 4.694444446

Combine like terms: -0.25 + 4.694444446 = 4.444444446
4.694444446 + -4.333333333x + x2 = 4.444444446

Factor a perfect square on the left side:
(x + -2.166666667)(x + -2.166666667) = 4.444444446

Calculate the square root of the right side: 2.108185107

Break this problem into two subproblems by setting 
(x + -2.166666667) equal to 2.108185107 and -2.108185107.


Subproblem 1
x + -2.166666667 = 2.108185107

Simplifying
x + -2.166666667 = 2.108185107

Reorder the terms:
-2.166666667 + x = 2.108185107

Solving
-2.166666667 + x = 2.108185107

Solving for variable 'x'.

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

Add '2.166666667' to each side of the equation.
-2.166666667 + 2.166666667 + x = 2.108185107 + 2.166666667

Combine like terms: -2.166666667 + 2.166666667 = 0.000000000
0.000000000 + x = 2.108185107 + 2.166666667
x = 2.108185107 + 2.166666667

Combine like terms: 2.108185107 + 2.166666667 = 4.274851774
x = 4.274851774

Simplifying
x = 4.274851774


Subproblem 2
x + -2.166666667 = -2.108185107

Simplifying
x + -2.166666667 = -2.108185107

Reorder the terms:
-2.166666667 + x = -2.108185107

Solving
-2.166666667 + x = -2.108185107

Solving for variable 'x'.

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

Add '2.166666667' to each side of the equation.
-2.166666667 + 2.166666667 + x = -2.108185107 + 2.166666667

Combine like terms: -2.166666667 + 2.166666667 = 0.000000000
0.000000000 + x = -2.108185107 + 2.166666667
x = -2.108185107 + 2.166666667

Combine like terms: -2.108185107 + 2.166666667 = 0.05848156
x = 0.05848156

Simplifying
x = 0.05848156


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {4.274851774, 0.05848156}

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