6x^2-42x+32=0

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Solution for 6x^2-42x+32=0 equation: 


Simplifying
6x2 + -42x + 32 = 0

Reorder the terms:
32 + -42x + 6x2 = 0

Solving
32 + -42x + 6x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '2'.
2(16 + -21x + 3x2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(16 + -21x + 3x2)' equal to zero and attempt to solve:

Simplifying
16 + -21x + 3x2 = 0

Solving
16 + -21x + 3x2 = 0

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

Divide each side by '3'.
5.333333333 + -7x + x2 = 0

Move the constant term to the right:

Add '-5.333333333' to each side of the equation.
5.333333333 + -7x + -5.333333333 + x2 = 0 + -5.333333333

Reorder the terms:
5.333333333 + -5.333333333 + -7x + x2 = 0 + -5.333333333

Combine like terms: 5.333333333 + -5.333333333 = 0.000000000
0.000000000 + -7x + x2 = 0 + -5.333333333
-7x + x2 = 0 + -5.333333333

Combine like terms: 0 + -5.333333333 = -5.333333333
-7x + x2 = -5.333333333

The x term is -7x.  Take half its coefficient (-3.5).
Square it (12.25) and add it to both sides.

Add '12.25' to each side of the equation.
-7x + 12.25 + x2 = -5.333333333 + 12.25

Reorder the terms:
12.25 + -7x + x2 = -5.333333333 + 12.25

Combine like terms: -5.333333333 + 12.25 = 6.916666667
12.25 + -7x + x2 = 6.916666667

Factor a perfect square on the left side:
(x + -3.5)(x + -3.5) = 6.916666667

Calculate the square root of the right side: 2.62995564

Break this problem into two subproblems by setting 
(x + -3.5) equal to 2.62995564 and -2.62995564.


Subproblem 1
x + -3.5 = 2.62995564

Simplifying
x + -3.5 = 2.62995564

Reorder the terms:
-3.5 + x = 2.62995564

Solving
-3.5 + x = 2.62995564

Solving for variable 'x'.

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

Add '3.5' to each side of the equation.
-3.5 + 3.5 + x = 2.62995564 + 3.5

Combine like terms: -3.5 + 3.5 = 0.0
0.0 + x = 2.62995564 + 3.5
x = 2.62995564 + 3.5

Combine like terms: 2.62995564 + 3.5 = 6.12995564
x = 6.12995564

Simplifying
x = 6.12995564


Subproblem 2
x + -3.5 = -2.62995564

Simplifying
x + -3.5 = -2.62995564

Reorder the terms:
-3.5 + x = -2.62995564

Solving
-3.5 + x = -2.62995564

Solving for variable 'x'.

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

Add '3.5' to each side of the equation.
-3.5 + 3.5 + x = -2.62995564 + 3.5

Combine like terms: -3.5 + 3.5 = 0.0
0.0 + x = -2.62995564 + 3.5
x = -2.62995564 + 3.5

Combine like terms: -2.62995564 + 3.5 = 0.87004436
x = 0.87004436

Simplifying
x = 0.87004436


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {6.12995564, 0.87004436}
Solution
x = {6.12995564, 0.87004436}

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