n^2-4n-1=0

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Solution for n^2-4n-1=0 equation:


Simplifying
n2 + -4n + -1 = 0

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

Solving
-1 + -4n + n2 = 0

Solving for variable 'n'.

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 = 5
4 + -4n + n2 = 5

Factor a perfect square on the left side:
(n + -2)(n + -2) = 5

Calculate the square root of the right side: 2.236067978

Break this problem into two subproblems by setting 
(n + -2) equal to 2.236067978 and -2.236067978.

Subproblem 1

n + -2 = 2.236067978 Simplifying n + -2 = 2.236067978 Reorder the terms: -2 + n = 2.236067978 Solving -2 + n = 2.236067978 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 = 2.236067978 + 2 Combine like terms: -2 + 2 = 0 0 + n = 2.236067978 + 2 n = 2.236067978 + 2 Combine like terms: 2.236067978 + 2 = 4.236067978 n = 4.236067978 Simplifying n = 4.236067978

Subproblem 2

n + -2 = -2.236067978 Simplifying n + -2 = -2.236067978 Reorder the terms: -2 + n = -2.236067978 Solving -2 + n = -2.236067978 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 = -2.236067978 + 2 Combine like terms: -2 + 2 = 0 0 + n = -2.236067978 + 2 n = -2.236067978 + 2 Combine like terms: -2.236067978 + 2 = -0.236067978 n = -0.236067978 Simplifying n = -0.236067978

Solution

The solution to the problem is based on the solutions from the subproblems. n = {4.236067978, -0.236067978}

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