0=x^2+4x-1

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


Simplifying
0 = x2 + 4x + -1

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

Solving
0 = -1 + 4x + x2

Solving for variable 'x'.

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

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

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

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

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

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 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

Subproblem 2

x + 2 = -2.236067978 Simplifying 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

Solution

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

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