6n(n+5)=66

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Solution for 6n(n+5)=66 equation: 


Simplifying
6n(n + 5) = 66

Reorder the terms:
6n(5 + n) = 66
(5 * 6n + n * 6n) = 66
(30n + 6n2) = 66

Solving
30n + 6n2 = 66

Solving for variable 'n'.

Reorder the terms:
-66 + 30n + 6n2 = 66 + -66

Combine like terms: 66 + -66 = 0
-66 + 30n + 6n2 = 0

Factor out the Greatest Common Factor (GCF), '6'.
6(-11 + 5n + n2) = 0

Ignore the factor 6.

Subproblem 1
Set the factor '(-11 + 5n + n2)' equal to zero and attempt to solve:

Simplifying
-11 + 5n + n2 = 0

Solving
-11 + 5n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '11' to each side of the equation.
-11 + 5n + 11 + n2 = 0 + 11

Reorder the terms:
-11 + 11 + 5n + n2 = 0 + 11

Combine like terms: -11 + 11 = 0
0 + 5n + n2 = 0 + 11
5n + n2 = 0 + 11

Combine like terms: 0 + 11 = 11
5n + n2 = 11

The n term is 5n.  Take half its coefficient (2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
5n + 6.25 + n2 = 11 + 6.25

Reorder the terms:
6.25 + 5n + n2 = 11 + 6.25

Combine like terms: 11 + 6.25 = 17.25
6.25 + 5n + n2 = 17.25

Factor a perfect square on the left side:
(n + 2.5)(n + 2.5) = 17.25

Calculate the square root of the right side: 4.153311931

Break this problem into two subproblems by setting 
(n + 2.5) equal to 4.153311931 and -4.153311931.


Subproblem 1
n + 2.5 = 4.153311931

Simplifying
n + 2.5 = 4.153311931

Reorder the terms:
2.5 + n = 4.153311931

Solving
2.5 + n = 4.153311931

Solving for variable 'n'.

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

Add '-2.5' to each side of the equation.
2.5 + -2.5 + n = 4.153311931 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + n = 4.153311931 + -2.5
n = 4.153311931 + -2.5

Combine like terms: 4.153311931 + -2.5 = 1.653311931
n = 1.653311931

Simplifying
n = 1.653311931


Subproblem 2
n + 2.5 = -4.153311931

Simplifying
n + 2.5 = -4.153311931

Reorder the terms:
2.5 + n = -4.153311931

Solving
2.5 + n = -4.153311931

Solving for variable 'n'.

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

Add '-2.5' to each side of the equation.
2.5 + -2.5 + n = -4.153311931 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + n = -4.153311931 + -2.5
n = -4.153311931 + -2.5

Combine like terms: -4.153311931 + -2.5 = -6.653311931
n = -6.653311931

Simplifying
n = -6.653311931


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {1.653311931, -6.653311931}
Solution
n = {1.653311931, -6.653311931}

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