2x+22=x^2+1

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Solution for 2x+22=x^2+1 equation:


Simplifying
2x + 22 = x2 + 1

Reorder the terms:
22 + 2x = x2 + 1

Reorder the terms:
22 + 2x = 1 + x2

Solving
22 + 2x = 1 + x2

Solving for variable 'x'.

Reorder the terms:
22 + -1 + 2x + -1x2 = 1 + x2 + -1 + -1x2

Combine like terms: 22 + -1 = 21
21 + 2x + -1x2 = 1 + x2 + -1 + -1x2

Reorder the terms:
21 + 2x + -1x2 = 1 + -1 + x2 + -1x2

Combine like terms: 1 + -1 = 0
21 + 2x + -1x2 = 0 + x2 + -1x2
21 + 2x + -1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
21 + 2x + -1x2 = 0

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

Divide each side by '-1'.
-21 + -2x + x2 = 0

Move the constant term to the right:

Add '21' to each side of the equation.
-21 + -2x + 21 + x2 = 0 + 21

Reorder the terms:
-21 + 21 + -2x + x2 = 0 + 21

Combine like terms: -21 + 21 = 0
0 + -2x + x2 = 0 + 21
-2x + x2 = 0 + 21

Combine like terms: 0 + 21 = 21
-2x + x2 = 21

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 = 21 + 1

Reorder the terms:
1 + -2x + x2 = 21 + 1

Combine like terms: 21 + 1 = 22
1 + -2x + x2 = 22

Factor a perfect square on the left side:
(x + -1)(x + -1) = 22

Calculate the square root of the right side: 4.69041576

Break this problem into two subproblems by setting 
(x + -1) equal to 4.69041576 and -4.69041576.

Subproblem 1

x + -1 = 4.69041576 Simplifying x + -1 = 4.69041576 Reorder the terms: -1 + x = 4.69041576 Solving -1 + x = 4.69041576 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 = 4.69041576 + 1 Combine like terms: -1 + 1 = 0 0 + x = 4.69041576 + 1 x = 4.69041576 + 1 Combine like terms: 4.69041576 + 1 = 5.69041576 x = 5.69041576 Simplifying x = 5.69041576

Subproblem 2

x + -1 = -4.69041576 Simplifying x + -1 = -4.69041576 Reorder the terms: -1 + x = -4.69041576 Solving -1 + x = -4.69041576 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 = -4.69041576 + 1 Combine like terms: -1 + 1 = 0 0 + x = -4.69041576 + 1 x = -4.69041576 + 1 Combine like terms: -4.69041576 + 1 = -3.69041576 x = -3.69041576 Simplifying x = -3.69041576

Solution

The solution to the problem is based on the solutions from the subproblems. x = {5.69041576, -3.69041576}

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