6x^2-216x+105=0

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Solution for 6x^2-216x+105=0 equation: 


Simplifying
6x2 + -216x + 105 = 0

Reorder the terms:
105 + -216x + 6x2 = 0

Solving
105 + -216x + 6x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '3'.
3(35 + -72x + 2x2) = 0

Ignore the factor 3.

Subproblem 1
Set the factor '(35 + -72x + 2x2)' equal to zero and attempt to solve:

Simplifying
35 + -72x + 2x2 = 0

Solving
35 + -72x + 2x2 = 0

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

Divide each side by '2'.
17.5 + -36x + x2 = 0

Move the constant term to the right:

Add '-17.5' to each side of the equation.
17.5 + -36x + -17.5 + x2 = 0 + -17.5

Reorder the terms:
17.5 + -17.5 + -36x + x2 = 0 + -17.5

Combine like terms: 17.5 + -17.5 = 0.0
0.0 + -36x + x2 = 0 + -17.5
-36x + x2 = 0 + -17.5

Combine like terms: 0 + -17.5 = -17.5
-36x + x2 = -17.5

The x term is -36x.  Take half its coefficient (-18).
Square it (324) and add it to both sides.

Add '324' to each side of the equation.
-36x + 324 + x2 = -17.5 + 324

Reorder the terms:
324 + -36x + x2 = -17.5 + 324

Combine like terms: -17.5 + 324 = 306.5
324 + -36x + x2 = 306.5

Factor a perfect square on the left side:
(x + -18)(x + -18) = 306.5

Calculate the square root of the right side: 17.5071414

Break this problem into two subproblems by setting 
(x + -18) equal to 17.5071414 and -17.5071414.


Subproblem 1
x + -18 = 17.5071414

Simplifying
x + -18 = 17.5071414

Reorder the terms:
-18 + x = 17.5071414

Solving
-18 + x = 17.5071414

Solving for variable 'x'.

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

Add '18' to each side of the equation.
-18 + 18 + x = 17.5071414 + 18

Combine like terms: -18 + 18 = 0
0 + x = 17.5071414 + 18
x = 17.5071414 + 18

Combine like terms: 17.5071414 + 18 = 35.5071414
x = 35.5071414

Simplifying
x = 35.5071414


Subproblem 2
x + -18 = -17.5071414

Simplifying
x + -18 = -17.5071414

Reorder the terms:
-18 + x = -17.5071414

Solving
-18 + x = -17.5071414

Solving for variable 'x'.

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

Add '18' to each side of the equation.
-18 + 18 + x = -17.5071414 + 18

Combine like terms: -18 + 18 = 0
0 + x = -17.5071414 + 18
x = -17.5071414 + 18

Combine like terms: -17.5071414 + 18 = 0.4928586
x = 0.4928586

Simplifying
x = 0.4928586


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {35.5071414, 0.4928586}
Solution
x = {35.5071414, 0.4928586}

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