(-x^2)+4x+1=0

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Solution for (-x^2)+4x+1=0 equation: 


Simplifying
(-1x2) + 4x + 1 = 0

Reorder the terms:
1 + 4x + (-1x2) = 0

Solving
1 + 4x + (-1x2) = 0

Solving for variable 'x'.

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

Divide each side by '-1'.
-1 + -4x + x2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -4x + 1 + x2 = 0 + 1

Reorder the terms:
-1 + 1 + -4x + x2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -4x + x2 = 0 + 1
-4x + x2 = 0 + 1

Combine like terms: 0 + 1 = 1
-4x + x2 = 1

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

Add '4' to each side of the equation.
-4x + 4 + x2 = 1 + 4

Reorder the terms:
4 + -4x + x2 = 1 + 4

Combine like terms: 1 + 4 = 5
4 + -4x + x2 = 5

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

Calculate the square root of the right side: 2.236067978

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


Subproblem 1
(x) + -2 = 2.236067978

Simplifying
(x) + -2 = 2.236067978
x + -2 = 2.236067978

Reorder the terms:
-2 + x = 2.236067978

Solving
-2 + x = 2.236067978

Solving for variable 'x'.

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

Add '2' to each side of the equation.
-2 + 2 + x = 2.236067978 + 2

Combine like terms: -2 + 2 = 0
0 + x = 2.236067978 + 2
x = 2.236067978 + 2

Combine like terms: 2.236067978 + 2 = 4.236067978
x = 4.236067978

Simplifying
x = 4.236067978


Subproblem 2
(x) + -2 = -2.236067978

Simplifying
(x) + -2 = -2.236067978
x + -2 = -2.236067978

Reorder the terms:
-2 + x = -2.236067978

Solving
-2 + x = -2.236067978

Solving for variable 'x'.

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

Add '2' to each side of the equation.
-2 + 2 + x = -2.236067978 + 2

Combine like terms: -2 + 2 = 0
0 + x = -2.236067978 + 2
x = -2.236067978 + 2

Combine like terms: -2.236067978 + 2 = -0.236067978
x = -0.236067978

Simplifying
x = -0.236067978


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

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