x+13=3x-8+x*x

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Solution for x+13=3x-8+x*x equation:


Simplifying
x + 13 = 3x + -8 + x * x

Reorder the terms:
13 + x = 3x + -8 + x * x

Multiply x * x
13 + x = 3x + -8 + x2

Reorder the terms:
13 + x = -8 + 3x + x2

Solving
13 + x = -8 + 3x + x2

Solving for variable 'x'.

Reorder the terms:
13 + 8 + x + -3x + -1x2 = -8 + 3x + x2 + 8 + -3x + -1x2

Combine like terms: 13 + 8 = 21
21 + x + -3x + -1x2 = -8 + 3x + x2 + 8 + -3x + -1x2

Combine like terms: x + -3x = -2x
21 + -2x + -1x2 = -8 + 3x + x2 + 8 + -3x + -1x2

Reorder the terms:
21 + -2x + -1x2 = -8 + 8 + 3x + -3x + x2 + -1x2

Combine like terms: -8 + 8 = 0
21 + -2x + -1x2 = 0 + 3x + -3x + x2 + -1x2
21 + -2x + -1x2 = 3x + -3x + x2 + -1x2

Combine like terms: 3x + -3x = 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 = 3.69041576 x = 3.69041576 Simplifying x = 3.69041576

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 = -5.69041576 x = -5.69041576 Simplifying x = -5.69041576

Solution

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

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