3x^2-32x+68=0

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Solution for 3x^2-32x+68=0 equation: 


Simplifying
3x2 + -32x + 68 = 0

Reorder the terms:
68 + -32x + 3x2 = 0

Solving
68 + -32x + 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'.
22.66666667 + -10.66666667x + x2 = 0

Move the constant term to the right:

Add '-22.66666667' to each side of the equation.
22.66666667 + -10.66666667x + -22.66666667 + x2 = 0 + -22.66666667

Reorder the terms:
22.66666667 + -22.66666667 + -10.66666667x + x2 = 0 + -22.66666667

Combine like terms: 22.66666667 + -22.66666667 = 0.00000000
0.00000000 + -10.66666667x + x2 = 0 + -22.66666667
-10.66666667x + x2 = 0 + -22.66666667

Combine like terms: 0 + -22.66666667 = -22.66666667
-10.66666667x + x2 = -22.66666667

The x term is -10.66666667x.  Take half its coefficient (-5.333333335).
Square it (28.44444446) and add it to both sides.

Add '28.44444446' to each side of the equation.
-10.66666667x + 28.44444446 + x2 = -22.66666667 + 28.44444446

Reorder the terms:
28.44444446 + -10.66666667x + x2 = -22.66666667 + 28.44444446

Combine like terms: -22.66666667 + 28.44444446 = 5.77777779
28.44444446 + -10.66666667x + x2 = 5.77777779

Factor a perfect square on the left side:
(x + -5.333333335)(x + -5.333333335) = 5.77777779

Calculate the square root of the right side: 2.403700853

Break this problem into two subproblems by setting 
(x + -5.333333335) equal to 2.403700853 and -2.403700853.


Subproblem 1
x + -5.333333335 = 2.403700853

Simplifying
x + -5.333333335 = 2.403700853

Reorder the terms:
-5.333333335 + x = 2.403700853

Solving
-5.333333335 + x = 2.403700853

Solving for variable 'x'.

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

Add '5.333333335' to each side of the equation.
-5.333333335 + 5.333333335 + x = 2.403700853 + 5.333333335

Combine like terms: -5.333333335 + 5.333333335 = 0.000000000
0.000000000 + x = 2.403700853 + 5.333333335
x = 2.403700853 + 5.333333335

Combine like terms: 2.403700853 + 5.333333335 = 7.737034188
x = 7.737034188

Simplifying
x = 7.737034188


Subproblem 2
x + -5.333333335 = -2.403700853

Simplifying
x + -5.333333335 = -2.403700853

Reorder the terms:
-5.333333335 + x = -2.403700853

Solving
-5.333333335 + x = -2.403700853

Solving for variable 'x'.

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

Add '5.333333335' to each side of the equation.
-5.333333335 + 5.333333335 + x = -2.403700853 + 5.333333335

Combine like terms: -5.333333335 + 5.333333335 = 0.000000000
0.000000000 + x = -2.403700853 + 5.333333335
x = -2.403700853 + 5.333333335

Combine like terms: -2.403700853 + 5.333333335 = 2.929632482
x = 2.929632482

Simplifying
x = 2.929632482


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {7.737034188, 2.929632482}

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