6x^2-18x+10=0

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Solution for 6x^2-18x+10=0 equation: 


Simplifying
6x2 + -18x + 10 = 0

Reorder the terms:
10 + -18x + 6x2 = 0

Solving
10 + -18x + 6x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '2'.
2(5 + -9x + 3x2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(5 + -9x + 3x2)' equal to zero and attempt to solve:

Simplifying
5 + -9x + 3x2 = 0

Solving
5 + -9x + 3x2 = 0

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

Divide each side by '3'.
1.666666667 + -3x + x2 = 0

Move the constant term to the right:

Add '-1.666666667' to each side of the equation.
1.666666667 + -3x + -1.666666667 + x2 = 0 + -1.666666667

Reorder the terms:
1.666666667 + -1.666666667 + -3x + x2 = 0 + -1.666666667

Combine like terms: 1.666666667 + -1.666666667 = 0.000000000
0.000000000 + -3x + x2 = 0 + -1.666666667
-3x + x2 = 0 + -1.666666667

Combine like terms: 0 + -1.666666667 = -1.666666667
-3x + x2 = -1.666666667

The x term is -3x.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3x + 2.25 + x2 = -1.666666667 + 2.25

Reorder the terms:
2.25 + -3x + x2 = -1.666666667 + 2.25

Combine like terms: -1.666666667 + 2.25 = 0.583333333
2.25 + -3x + x2 = 0.583333333

Factor a perfect square on the left side:
(x + -1.5)(x + -1.5) = 0.583333333

Calculate the square root of the right side: 0.763762616

Break this problem into two subproblems by setting 
(x + -1.5) equal to 0.763762616 and -0.763762616.


Subproblem 1
x + -1.5 = 0.763762616

Simplifying
x + -1.5 = 0.763762616

Reorder the terms:
-1.5 + x = 0.763762616

Solving
-1.5 + x = 0.763762616

Solving for variable 'x'.

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

Add '1.5' to each side of the equation.
-1.5 + 1.5 + x = 0.763762616 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + x = 0.763762616 + 1.5
x = 0.763762616 + 1.5

Combine like terms: 0.763762616 + 1.5 = 2.263762616
x = 2.263762616

Simplifying
x = 2.263762616


Subproblem 2
x + -1.5 = -0.763762616

Simplifying
x + -1.5 = -0.763762616

Reorder the terms:
-1.5 + x = -0.763762616

Solving
-1.5 + x = -0.763762616

Solving for variable 'x'.

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

Add '1.5' to each side of the equation.
-1.5 + 1.5 + x = -0.763762616 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + x = -0.763762616 + 1.5
x = -0.763762616 + 1.5

Combine like terms: -0.763762616 + 1.5 = 0.736237384
x = 0.736237384

Simplifying
x = 0.736237384


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.263762616, 0.736237384}
Solution
x = {2.263762616, 0.736237384}

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