6(n-3)n=10

Solution for 6(n-3)n=10 equation:

Simplifying
6(n + -3) * n = 10

Reorder the terms:
6(-3 + n) * n = 10

Reorder the terms for easier multiplication:
6n(-3 + n) = 10
(-3 * 6n + n * 6n) = 10
(-18n + 6n2) = 10

Solving
-18n + 6n2 = 10

Solving for variable 'n'.

Reorder the terms:
-10 + -18n + 6n2 = 10 + -10

Combine like terms: 10 + -10 = 0
-10 + -18n + 6n2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-5 + -9n + 3n2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-5 + -9n + 3n2)' equal to zero and attempt to solve:

Simplifying
-5 + -9n + 3n2 = 0

Solving
-5 + -9n + 3n2 = 0

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

Divide each side by '3'.
-1.666666667 + -3n + n2 = 0

Move the constant term to the right:

Add '1.666666667' to each side of the equation.
-1.666666667 + -3n + 1.666666667 + n2 = 0 + 1.666666667

Reorder the terms:
-1.666666667 + 1.666666667 + -3n + n2 = 0 + 1.666666667

Combine like terms: -1.666666667 + 1.666666667 = 0.000000000
0.000000000 + -3n + n2 = 0 + 1.666666667
-3n + n2 = 0 + 1.666666667

Combine like terms: 0 + 1.666666667 = 1.666666667
-3n + n2 = 1.666666667

The n term is -3n.  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.
-3n + 2.25 + n2 = 1.666666667 + 2.25

Reorder the terms:
2.25 + -3n + n2 = 1.666666667 + 2.25

Combine like terms: 1.666666667 + 2.25 = 3.916666667
2.25 + -3n + n2 = 3.916666667

Factor a perfect square on the left side:
(n + -1.5)(n + -1.5) = 3.916666667

Calculate the square root of the right side: 1.979057015

Break this problem into two subproblems by setting 
(n + -1.5) equal to 1.979057015 and -1.979057015.

Subproblem 1
n + -1.5 = 1.979057015

Simplifying
n + -1.5 = 1.979057015

Reorder the terms:
-1.5 + n = 1.979057015

Solving
-1.5 + n = 1.979057015

Solving for variable 'n'.

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

Add '1.5' to each side of the equation.
-1.5 + 1.5 + n = 1.979057015 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + n = 1.979057015 + 1.5
n = 1.979057015 + 1.5

Combine like terms: 1.979057015 + 1.5 = 3.479057015
n = 3.479057015

Simplifying
n = 3.479057015

Subproblem 2
n + -1.5 = -1.979057015

Simplifying
n + -1.5 = -1.979057015

Reorder the terms:
-1.5 + n = -1.979057015

Solving
-1.5 + n = -1.979057015

Solving for variable 'n'.

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

Add '1.5' to each side of the equation.
-1.5 + 1.5 + n = -1.979057015 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + n = -1.979057015 + 1.5
n = -1.979057015 + 1.5

Combine like terms: -1.979057015 + 1.5 = -0.479057015
n = -0.479057015

Simplifying
n = -0.479057015

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {3.479057015, -0.479057015}
Solution
n = {3.479057015, -0.479057015}