3(n+11)-5n=-2n(n-12)+19

Solution for 3(n+11)-5n=-2n(n-12)+19 equation:

Simplifying
3(n + 11) + -5n = -2n(n + -12) + 19

Reorder the terms:
3(11 + n) + -5n = -2n(n + -12) + 19
(11 * 3 + n * 3) + -5n = -2n(n + -12) + 19
(33 + 3n) + -5n = -2n(n + -12) + 19

Combine like terms: 3n + -5n = -2n
33 + -2n = -2n(n + -12) + 19

Reorder the terms:
33 + -2n = -2n(-12 + n) + 19
33 + -2n = (-12 * -2n + n * -2n) + 19
33 + -2n = (24n + -2n2) + 19

Reorder the terms:
33 + -2n = 19 + 24n + -2n2

Solving
33 + -2n = 19 + 24n + -2n2

Solving for variable 'n'.

Reorder the terms:
33 + -19 + -2n + -24n + 2n2 = 19 + 24n + -2n2 + -19 + -24n + 2n2

Combine like terms: 33 + -19 = 14
14 + -2n + -24n + 2n2 = 19 + 24n + -2n2 + -19 + -24n + 2n2

Combine like terms: -2n + -24n = -26n
14 + -26n + 2n2 = 19 + 24n + -2n2 + -19 + -24n + 2n2

Reorder the terms:
14 + -26n + 2n2 = 19 + -19 + 24n + -24n + -2n2 + 2n2

Combine like terms: 19 + -19 = 0
14 + -26n + 2n2 = 0 + 24n + -24n + -2n2 + 2n2
14 + -26n + 2n2 = 24n + -24n + -2n2 + 2n2

Combine like terms: 24n + -24n = 0
14 + -26n + 2n2 = 0 + -2n2 + 2n2
14 + -26n + 2n2 = -2n2 + 2n2

Combine like terms: -2n2 + 2n2 = 0
14 + -26n + 2n2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(7 + -13n + n2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(7 + -13n + n2)' equal to zero and attempt to solve:

Simplifying
7 + -13n + n2 = 0

Solving
7 + -13n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-7' to each side of the equation.
7 + -13n + -7 + n2 = 0 + -7

Reorder the terms:
7 + -7 + -13n + n2 = 0 + -7

Combine like terms: 7 + -7 = 0
0 + -13n + n2 = 0 + -7
-13n + n2 = 0 + -7

Combine like terms: 0 + -7 = -7
-13n + n2 = -7

The n term is -13n.  Take half its coefficient (-6.5).
Square it (42.25) and add it to both sides.

Add '42.25' to each side of the equation.
-13n + 42.25 + n2 = -7 + 42.25

Reorder the terms:
42.25 + -13n + n2 = -7 + 42.25

Combine like terms: -7 + 42.25 = 35.25
42.25 + -13n + n2 = 35.25

Factor a perfect square on the left side:
(n + -6.5)(n + -6.5) = 35.25

Calculate the square root of the right side: 5.937171044

Break this problem into two subproblems by setting 
(n + -6.5) equal to 5.937171044 and -5.937171044.

Subproblem 1
n + -6.5 = 5.937171044

Simplifying
n + -6.5 = 5.937171044

Reorder the terms:
-6.5 + n = 5.937171044

Solving
-6.5 + n = 5.937171044

Solving for variable 'n'.

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

Add '6.5' to each side of the equation.
-6.5 + 6.5 + n = 5.937171044 + 6.5

Combine like terms: -6.5 + 6.5 = 0.0
0.0 + n = 5.937171044 + 6.5
n = 5.937171044 + 6.5

Combine like terms: 5.937171044 + 6.5 = 12.437171044
n = 12.437171044

Simplifying
n = 12.437171044

Subproblem 2
n + -6.5 = -5.937171044

Simplifying
n + -6.5 = -5.937171044

Reorder the terms:
-6.5 + n = -5.937171044

Solving
-6.5 + n = -5.937171044

Solving for variable 'n'.

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

Add '6.5' to each side of the equation.
-6.5 + 6.5 + n = -5.937171044 + 6.5

Combine like terms: -6.5 + 6.5 = 0.0
0.0 + n = -5.937171044 + 6.5
n = -5.937171044 + 6.5

Combine like terms: -5.937171044 + 6.5 = 0.562828956
n = 0.562828956

Simplifying
n = 0.562828956

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {12.437171044, 0.562828956}
Solution
n = {12.437171044, 0.562828956}