x+1=x^2-6

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


Simplifying
x + 1 = x2 + -6

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

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

Solving
1 + x = -6 + x2

Solving for variable 'x'.

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

Combine like terms: 1 + 6 = 7
7 + x + -1x2 = -6 + x2 + 6 + -1x2

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

Combine like terms: -6 + 6 = 0
7 + x + -1x2 = 0 + x2 + -1x2
7 + x + -1x2 = x2 + -1x2

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

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

Divide each side by '-1'.
-7 + -1x + x2 = 0

Move the constant term to the right:

Add '7' to each side of the equation.
-7 + -1x + 7 + x2 = 0 + 7

Reorder the terms:
-7 + 7 + -1x + x2 = 0 + 7

Combine like terms: -7 + 7 = 0
0 + -1x + x2 = 0 + 7
-1x + x2 = 0 + 7

Combine like terms: 0 + 7 = 7
-1x + x2 = 7

The x term is x.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1x + 0.25 + x2 = 7 + 0.25

Reorder the terms:
0.25 + -1x + x2 = 7 + 0.25

Combine like terms: 7 + 0.25 = 7.25
0.25 + -1x + x2 = 7.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 7.25

Calculate the square root of the right side: 2.692582404

Break this problem into two subproblems by setting 
(x + 0.5) equal to 2.692582404 and -2.692582404.

Subproblem 1

x + 0.5 = 2.692582404 Simplifying x + 0.5 = 2.692582404 Reorder the terms: 0.5 + x = 2.692582404 Solving 0.5 + x = 2.692582404 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = 2.692582404 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 2.692582404 + -0.5 x = 2.692582404 + -0.5 Combine like terms: 2.692582404 + -0.5 = 2.192582404 x = 2.192582404 Simplifying x = 2.192582404

Subproblem 2

x + 0.5 = -2.692582404 Simplifying x + 0.5 = -2.692582404 Reorder the terms: 0.5 + x = -2.692582404 Solving 0.5 + x = -2.692582404 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = -2.692582404 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -2.692582404 + -0.5 x = -2.692582404 + -0.5 Combine like terms: -2.692582404 + -0.5 = -3.192582404 x = -3.192582404 Simplifying x = -3.192582404

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.192582404, -3.192582404}

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