n^2=-4n-1

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Solution for n^2=-4n-1 equation:


Simplifying
n2 = -4n + -1

Reorder the terms:
n2 = -1 + -4n

Solving
n2 = -1 + -4n

Solving for variable 'n'.

Reorder the terms:
1 + 4n + n2 = -1 + -4n + 1 + 4n

Reorder the terms:
1 + 4n + n2 = -1 + 1 + -4n + 4n

Combine like terms: -1 + 1 = 0
1 + 4n + n2 = 0 + -4n + 4n
1 + 4n + n2 = -4n + 4n

Combine like terms: -4n + 4n = 0
1 + 4n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + 4n + -1 + n2 = 0 + -1

Reorder the terms:
1 + -1 + 4n + n2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + 4n + n2 = 0 + -1
4n + n2 = 0 + -1

Combine like terms: 0 + -1 = -1
4n + n2 = -1

The n term is 4n.  Take half its coefficient (2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
4n + 4 + n2 = -1 + 4

Reorder the terms:
4 + 4n + n2 = -1 + 4

Combine like terms: -1 + 4 = 3
4 + 4n + n2 = 3

Factor a perfect square on the left side:
(n + 2)(n + 2) = 3

Calculate the square root of the right side: 1.732050808

Break this problem into two subproblems by setting 
(n + 2) equal to 1.732050808 and -1.732050808.

Subproblem 1

n + 2 = 1.732050808 Simplifying n + 2 = 1.732050808 Reorder the terms: 2 + n = 1.732050808 Solving 2 + n = 1.732050808 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + n = 1.732050808 + -2 Combine like terms: 2 + -2 = 0 0 + n = 1.732050808 + -2 n = 1.732050808 + -2 Combine like terms: 1.732050808 + -2 = -0.267949192 n = -0.267949192 Simplifying n = -0.267949192

Subproblem 2

n + 2 = -1.732050808 Simplifying n + 2 = -1.732050808 Reorder the terms: 2 + n = -1.732050808 Solving 2 + n = -1.732050808 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + n = -1.732050808 + -2 Combine like terms: 2 + -2 = 0 0 + n = -1.732050808 + -2 n = -1.732050808 + -2 Combine like terms: -1.732050808 + -2 = -3.732050808 n = -3.732050808 Simplifying n = -3.732050808

Solution

The solution to the problem is based on the solutions from the subproblems. n = {-0.267949192, -3.732050808}

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