x+6=x^2-x+1

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Solution for x+6=x^2-x+1 equation:


Simplifying
x + 6 = x2 + -1x + 1

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

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

Solving
6 + x = 1 + -1x + x2

Solving for variable 'x'.

Reorder the terms:
6 + -1 + x + x + -1x2 = 1 + -1x + x2 + -1 + x + -1x2

Combine like terms: 6 + -1 = 5
5 + x + x + -1x2 = 1 + -1x + x2 + -1 + x + -1x2

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

Reorder the terms:
5 + 2x + -1x2 = 1 + -1 + -1x + x + x2 + -1x2

Combine like terms: 1 + -1 = 0
5 + 2x + -1x2 = 0 + -1x + x + x2 + -1x2
5 + 2x + -1x2 = -1x + x + x2 + -1x2

Combine like terms: -1x + x = 0
5 + 2x + -1x2 = 0 + x2 + -1x2
5 + 2x + -1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
5 + 2x + -1x2 = 0

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

Divide each side by '-1'.
-5 + -2x + x2 = 0

Move the constant term to the right:

Add '5' to each side of the equation.
-5 + -2x + 5 + x2 = 0 + 5

Reorder the terms:
-5 + 5 + -2x + x2 = 0 + 5

Combine like terms: -5 + 5 = 0
0 + -2x + x2 = 0 + 5
-2x + x2 = 0 + 5

Combine like terms: 0 + 5 = 5
-2x + x2 = 5

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 = 5 + 1

Reorder the terms:
1 + -2x + x2 = 5 + 1

Combine like terms: 5 + 1 = 6
1 + -2x + x2 = 6

Factor a perfect square on the left side:
(x + -1)(x + -1) = 6

Calculate the square root of the right side: 2.449489743

Break this problem into two subproblems by setting 
(x + -1) equal to 2.449489743 and -2.449489743.

Subproblem 1

x + -1 = 2.449489743 Simplifying x + -1 = 2.449489743 Reorder the terms: -1 + x = 2.449489743 Solving -1 + x = 2.449489743 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 = 2.449489743 + 1 Combine like terms: -1 + 1 = 0 0 + x = 2.449489743 + 1 x = 2.449489743 + 1 Combine like terms: 2.449489743 + 1 = 3.449489743 x = 3.449489743 Simplifying x = 3.449489743

Subproblem 2

x + -1 = -2.449489743 Simplifying x + -1 = -2.449489743 Reorder the terms: -1 + x = -2.449489743 Solving -1 + x = -2.449489743 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 = -2.449489743 + 1 Combine like terms: -1 + 1 = 0 0 + x = -2.449489743 + 1 x = -2.449489743 + 1 Combine like terms: -2.449489743 + 1 = -1.449489743 x = -1.449489743 Simplifying x = -1.449489743

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.449489743, -1.449489743}

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