5-x=x^2+x+1

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Solution for 5-x=x^2+x+1 equation:


Simplifying
5 + -1x = x2 + x + 1

Reorder the terms:
5 + -1x = 1 + x + x2

Solving
5 + -1x = 1 + x + x2

Solving for variable 'x'.

Reorder the terms:
5 + -1 + -1x + -1x + -1x2 = 1 + x + x2 + -1 + -1x + -1x2

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

Combine like terms: -1x + -1x = -2x
4 + -2x + -1x2 = 1 + x + x2 + -1 + -1x + -1x2

Reorder the terms:
4 + -2x + -1x2 = 1 + -1 + x + -1x + x2 + -1x2

Combine like terms: 1 + -1 = 0
4 + -2x + -1x2 = 0 + x + -1x + x2 + -1x2
4 + -2x + -1x2 = x + -1x + x2 + -1x2

Combine like terms: x + -1x = 0
4 + -2x + -1x2 = 0 + x2 + -1x2
4 + -2x + -1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
4 + -2x + -1x2 = 0

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

Divide each side by '-1'.
-4 + 2x + x2 = 0

Move the constant term to the right:

Add '4' to each side of the equation.
-4 + 2x + 4 + x2 = 0 + 4

Reorder the terms:
-4 + 4 + 2x + x2 = 0 + 4

Combine like terms: -4 + 4 = 0
0 + 2x + x2 = 0 + 4
2x + x2 = 0 + 4

Combine like terms: 0 + 4 = 4
2x + x2 = 4

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

Add '1' to each side of the equation.
2x + 1 + x2 = 4 + 1

Reorder the terms:
1 + 2x + x2 = 4 + 1

Combine like terms: 4 + 1 = 5
1 + 2x + x2 = 5

Factor a perfect square on the left side:
(x + 1)(x + 1) = 5

Calculate the square root of the right side: 2.236067978

Break this problem into two subproblems by setting 
(x + 1) equal to 2.236067978 and -2.236067978.

Subproblem 1

x + 1 = 2.236067978 Simplifying x + 1 = 2.236067978 Reorder the terms: 1 + x = 2.236067978 Solving 1 + x = 2.236067978 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + x = 2.236067978 + -1 Combine like terms: 1 + -1 = 0 0 + x = 2.236067978 + -1 x = 2.236067978 + -1 Combine like terms: 2.236067978 + -1 = 1.236067978 x = 1.236067978 Simplifying x = 1.236067978

Subproblem 2

x + 1 = -2.236067978 Simplifying x + 1 = -2.236067978 Reorder the terms: 1 + x = -2.236067978 Solving 1 + x = -2.236067978 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + x = -2.236067978 + -1 Combine like terms: 1 + -1 = 0 0 + x = -2.236067978 + -1 x = -2.236067978 + -1 Combine like terms: -2.236067978 + -1 = -3.236067978 x = -3.236067978 Simplifying x = -3.236067978

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.236067978, -3.236067978}

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