6(N-3)34N=16

Solution for 6(N-3)34N=16 equation:

Simplifying
6(N + -3) * 34N = 16

Reorder the terms:
6(-3 + N) * 34N = 16

Reorder the terms for easier multiplication:
6 * 34N(-3 + N) = 16

Multiply 6 * 34
204N(-3 + N) = 16
(-3 * 204N + N * 204N) = 16
(-612N + 204N2) = 16

Solving
-612N + 204N2 = 16

Solving for variable 'N'.

Reorder the terms:
-16 + -612N + 204N2 = 16 + -16

Combine like terms: 16 + -16 = 0
-16 + -612N + 204N2 = 0

Factor out the Greatest Common Factor (GCF), '4'.
4(-4 + -153N + 51N2) = 0

Ignore the factor 4.
Subproblem 1
Set the factor '(-4 + -153N + 51N2)' equal to zero and attempt to solve:

Simplifying
-4 + -153N + 51N2 = 0

Solving
-4 + -153N + 51N2 = 0

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

Divide each side by '51'.
-0.07843137255 + -3N + N2 = 0

Move the constant term to the right:

Add '0.07843137255' to each side of the equation.
-0.07843137255 + -3N + 0.07843137255 + N2 = 0 + 0.07843137255

Reorder the terms:
-0.07843137255 + 0.07843137255 + -3N + N2 = 0 + 0.07843137255

Combine like terms: -0.07843137255 + 0.07843137255 = 0.00000000000
0.00000000000 + -3N + N2 = 0 + 0.07843137255
-3N + N2 = 0 + 0.07843137255

Combine like terms: 0 + 0.07843137255 = 0.07843137255
-3N + N2 = 0.07843137255

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 = 0.07843137255 + 2.25

Reorder the terms:
2.25 + -3N + N2 = 0.07843137255 + 2.25

Combine like terms: 0.07843137255 + 2.25 = 2.32843137255
2.25 + -3N + N2 = 2.32843137255

Factor a perfect square on the left side:
(N + -1.5)(N + -1.5) = 2.32843137255

Calculate the square root of the right side: 1.525919845

Break this problem into two subproblems by setting 
(N + -1.5) equal to 1.525919845 and -1.525919845.

Subproblem 1
N + -1.5 = 1.525919845

Simplifying
N + -1.5 = 1.525919845

Reorder the terms:
-1.5 + N = 1.525919845

Solving
-1.5 + N = 1.525919845

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 = 1.525919845 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + N = 1.525919845 + 1.5
N = 1.525919845 + 1.5

Combine like terms: 1.525919845 + 1.5 = 3.025919845
N = 3.025919845

Simplifying
N = 3.025919845

Subproblem 2
N + -1.5 = -1.525919845

Simplifying
N + -1.5 = -1.525919845

Reorder the terms:
-1.5 + N = -1.525919845

Solving
-1.5 + N = -1.525919845

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 = -1.525919845 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + N = -1.525919845 + 1.5
N = -1.525919845 + 1.5

Combine like terms: -1.525919845 + 1.5 = -0.025919845
N = -0.025919845

Simplifying
N = -0.025919845

Solution
The solution to the problem is based on the solutions
from the subproblems.
N = {3.025919845, -0.025919845}
Solution
N = {3.025919845, -0.025919845}