x^2+1=42-x

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Solution for x^2+1=42-x equation:


Simplifying
x2 + 1 = 42 + -1x

Reorder the terms:
1 + x2 = 42 + -1x

Solving
1 + x2 = 42 + -1x

Solving for variable 'x'.

Reorder the terms:
1 + -42 + x + x2 = 42 + -1x + -42 + x

Combine like terms: 1 + -42 = -41
-41 + x + x2 = 42 + -1x + -42 + x

Reorder the terms:
-41 + x + x2 = 42 + -42 + -1x + x

Combine like terms: 42 + -42 = 0
-41 + x + x2 = 0 + -1x + x
-41 + x + x2 = -1x + x

Combine like terms: -1x + x = 0
-41 + x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '41' to each side of the equation.
-41 + x + 41 + x2 = 0 + 41

Reorder the terms:
-41 + 41 + x + x2 = 0 + 41

Combine like terms: -41 + 41 = 0
0 + x + x2 = 0 + 41
x + x2 = 0 + 41

Combine like terms: 0 + 41 = 41
x + x2 = 41

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.
x + 0.25 + x2 = 41 + 0.25

Reorder the terms:
0.25 + x + x2 = 41 + 0.25

Combine like terms: 41 + 0.25 = 41.25
0.25 + x + x2 = 41.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 41.25

Calculate the square root of the right side: 6.422616289

Break this problem into two subproblems by setting 
(x + 0.5) equal to 6.422616289 and -6.422616289.

Subproblem 1

x + 0.5 = 6.422616289 Simplifying x + 0.5 = 6.422616289 Reorder the terms: 0.5 + x = 6.422616289 Solving 0.5 + x = 6.422616289 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 = 6.422616289 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 6.422616289 + -0.5 x = 6.422616289 + -0.5 Combine like terms: 6.422616289 + -0.5 = 5.922616289 x = 5.922616289 Simplifying x = 5.922616289

Subproblem 2

x + 0.5 = -6.422616289 Simplifying x + 0.5 = -6.422616289 Reorder the terms: 0.5 + x = -6.422616289 Solving 0.5 + x = -6.422616289 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 = -6.422616289 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -6.422616289 + -0.5 x = -6.422616289 + -0.5 Combine like terms: -6.422616289 + -0.5 = -6.922616289 x = -6.922616289 Simplifying x = -6.922616289

Solution

The solution to the problem is based on the solutions from the subproblems. x = {5.922616289, -6.922616289}

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