5n^2=-10n-2

Solution for 5n^2=-10n-2 equation:

Simplifying
5n2 = -10n + -2

Reorder the terms:
5n2 = -2 + -10n

Solving
5n2 = -2 + -10n

Solving for variable 'n'.

Reorder the terms:
2 + 10n + 5n2 = -2 + -10n + 2 + 10n

Reorder the terms:
2 + 10n + 5n2 = -2 + 2 + -10n + 10n

Combine like terms: -2 + 2 = 0
2 + 10n + 5n2 = 0 + -10n + 10n
2 + 10n + 5n2 = -10n + 10n

Combine like terms: -10n + 10n = 0
2 + 10n + 5n2 = 0

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

Divide each side by '5'.
0.4 + 2n + n2 = 0

Move the constant term to the right:

Add '-0.4' to each side of the equation.
0.4 + 2n + -0.4 + n2 = 0 + -0.4

Reorder the terms:
0.4 + -0.4 + 2n + n2 = 0 + -0.4

Combine like terms: 0.4 + -0.4 = 0.0
0.0 + 2n + n2 = 0 + -0.4
2n + n2 = 0 + -0.4

Combine like terms: 0 + -0.4 = -0.4
2n + n2 = -0.4

The n term is 2n.  Take half its coefficient (1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
2n + 1 + n2 = -0.4 + 1

Reorder the terms:
1 + 2n + n2 = -0.4 + 1

Combine like terms: -0.4 + 1 = 0.6
1 + 2n + n2 = 0.6

Factor a perfect square on the left side:
(n + 1)(n + 1) = 0.6

Calculate the square root of the right side: 0.774596669

Break this problem into two subproblems by setting 
(n + 1) equal to 0.774596669 and -0.774596669.

Subproblem 1
n + 1 = 0.774596669

Simplifying
n + 1 = 0.774596669

Reorder the terms:
1 + n = 0.774596669

Solving
1 + n = 0.774596669

Solving for variable 'n'.

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

Add '-1' to each side of the equation.
1 + -1 + n = 0.774596669 + -1

Combine like terms: 1 + -1 = 0
0 + n = 0.774596669 + -1
n = 0.774596669 + -1

Combine like terms: 0.774596669 + -1 = -0.225403331
n = -0.225403331

Simplifying
n = -0.225403331

Subproblem 2
n + 1 = -0.774596669

Simplifying
n + 1 = -0.774596669

Reorder the terms:
1 + n = -0.774596669

Solving
1 + n = -0.774596669

Solving for variable 'n'.

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

Add '-1' to each side of the equation.
1 + -1 + n = -0.774596669 + -1

Combine like terms: 1 + -1 = 0
0 + n = -0.774596669 + -1
n = -0.774596669 + -1

Combine like terms: -0.774596669 + -1 = -1.774596669
n = -1.774596669

Simplifying
n = -1.774596669

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {-0.225403331, -1.774596669}