1-3n(n+5)=-8

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Solution for 1-3n(n+5)=-8 equation: 


Simplifying
1 + -3n(n + 5) = -8

Reorder the terms:
1 + -3n(5 + n) = -8
1 + (5 * -3n + n * -3n) = -8
1 + (-15n + -3n2) = -8

Solving
1 + -15n + -3n2 = -8

Solving for variable 'n'.

Reorder the terms:
1 + 8 + -15n + -3n2 = -8 + 8

Combine like terms: 1 + 8 = 9
9 + -15n + -3n2 = -8 + 8

Combine like terms: -8 + 8 = 0
9 + -15n + -3n2 = 0

Factor out the Greatest Common Factor (GCF), '3'.
3(3 + -5n + -1n2) = 0

Ignore the factor 3.

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

Simplifying
3 + -5n + -1n2 = 0

Solving
3 + -5n + -1n2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-3 + 5n + n2 = 0

Move the constant term to the right:

Add '3' to each side of the equation.
-3 + 5n + 3 + n2 = 0 + 3

Reorder the terms:
-3 + 3 + 5n + n2 = 0 + 3

Combine like terms: -3 + 3 = 0
0 + 5n + n2 = 0 + 3
5n + n2 = 0 + 3

Combine like terms: 0 + 3 = 3
5n + n2 = 3

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 = 3 + 6.25

Reorder the terms:
6.25 + 5n + n2 = 3 + 6.25

Combine like terms: 3 + 6.25 = 9.25
6.25 + 5n + n2 = 9.25

Factor a perfect square on the left side:
(n + 2.5)(n + 2.5) = 9.25

Calculate the square root of the right side: 3.041381265

Break this problem into two subproblems by setting 
(n + 2.5) equal to 3.041381265 and -3.041381265.


Subproblem 1
n + 2.5 = 3.041381265

Simplifying
n + 2.5 = 3.041381265

Reorder the terms:
2.5 + n = 3.041381265

Solving
2.5 + n = 3.041381265

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 = 3.041381265 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + n = 3.041381265 + -2.5
n = 3.041381265 + -2.5

Combine like terms: 3.041381265 + -2.5 = 0.541381265
n = 0.541381265

Simplifying
n = 0.541381265


Subproblem 2
n + 2.5 = -3.041381265

Simplifying
n + 2.5 = -3.041381265

Reorder the terms:
2.5 + n = -3.041381265

Solving
2.5 + n = -3.041381265

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 = -3.041381265 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + n = -3.041381265 + -2.5
n = -3.041381265 + -2.5

Combine like terms: -3.041381265 + -2.5 = -5.541381265
n = -5.541381265

Simplifying
n = -5.541381265


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {0.541381265, -5.541381265}
Solution
n = {0.541381265, -5.541381265}

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