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

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

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

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

Multiply n * n
15 + 5n = 3(n2 + -1)

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

Solving
15 + 5n = -3 + 3n2

Solving for variable 'n'.

Reorder the terms:
15 + 3 + 5n + -3n2 = -3 + 3n2 + 3 + -3n2

Combine like terms: 15 + 3 = 18
18 + 5n + -3n2 = -3 + 3n2 + 3 + -3n2

Reorder the terms:
18 + 5n + -3n2 = -3 + 3 + 3n2 + -3n2

Combine like terms: -3 + 3 = 0
18 + 5n + -3n2 = 0 + 3n2 + -3n2
18 + 5n + -3n2 = 3n2 + -3n2

Combine like terms: 3n2 + -3n2 = 0
18 + 5n + -3n2 = 0

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

Divide each side by '-3'.
-6 + -1.666666667n + n2 = 0

Move the constant term to the right:

Add '6' to each side of the equation.
-6 + -1.666666667n + 6 + n2 = 0 + 6

Reorder the terms:
-6 + 6 + -1.666666667n + n2 = 0 + 6

Combine like terms: -6 + 6 = 0
0 + -1.666666667n + n2 = 0 + 6
-1.666666667n + n2 = 0 + 6

Combine like terms: 0 + 6 = 6
-1.666666667n + n2 = 6

The n term is -1.666666667n.  Take half its coefficient (-0.8333333335).
Square it (0.6944444447) and add it to both sides.

Add '0.6944444447' to each side of the equation.
-1.666666667n + 0.6944444447 + n2 = 6 + 0.6944444447

Reorder the terms:
0.6944444447 + -1.666666667n + n2 = 6 + 0.6944444447

Combine like terms: 6 + 0.6944444447 = 6.6944444447
0.6944444447 + -1.666666667n + n2 = 6.6944444447

Factor a perfect square on the left side:
(n + -0.8333333335)(n + -0.8333333335) = 6.6944444447

Calculate the square root of the right side: 2.587362449

Break this problem into two subproblems by setting 
(n + -0.8333333335) equal to 2.587362449 and -2.587362449.

Subproblem 1
n + -0.8333333335 = 2.587362449

Simplifying
n + -0.8333333335 = 2.587362449

Reorder the terms:
-0.8333333335 + n = 2.587362449

Solving
-0.8333333335 + n = 2.587362449

Solving for variable 'n'.

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

Add '0.8333333335' to each side of the equation.
-0.8333333335 + 0.8333333335 + n = 2.587362449 + 0.8333333335

Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000
0.0000000000 + n = 2.587362449 + 0.8333333335
n = 2.587362449 + 0.8333333335

Combine like terms: 2.587362449 + 0.8333333335 = 3.4206957825
n = 3.4206957825

Simplifying
n = 3.4206957825

Subproblem 2
n + -0.8333333335 = -2.587362449

Simplifying
n + -0.8333333335 = -2.587362449

Reorder the terms:
-0.8333333335 + n = -2.587362449

Solving
-0.8333333335 + n = -2.587362449

Solving for variable 'n'.

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

Add '0.8333333335' to each side of the equation.
-0.8333333335 + 0.8333333335 + n = -2.587362449 + 0.8333333335

Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000
0.0000000000 + n = -2.587362449 + 0.8333333335
n = -2.587362449 + 0.8333333335

Combine like terms: -2.587362449 + 0.8333333335 = -1.7540291155
n = -1.7540291155

Simplifying
n = -1.7540291155

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {3.4206957825, -1.7540291155}