x(2+x)=8+x

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Solution for x(2+x)=8+x equation:


Simplifying
x(2 + x) = 8 + x
(2 * x + x * x) = 8 + x
(2x + x2) = 8 + x

Solving
2x + x2 = 8 + x

Solving for variable 'x'.

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

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

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

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

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

Begin completing the square.

Move the constant term to the right:

Add '8' to each side of the equation.
-8 + 1x + 8 + x2 = 0 + 8

Reorder the terms:
-8 + 8 + 1x + x2 = 0 + 8

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

Combine like terms: 0 + 8 = 8
1x + x2 = 8

The x term is 1x.  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 = 8 + 0.25

Reorder the terms:
0.25 + x + x2 = 8 + 0.25

Combine like terms: 8 + 0.25 = 8.25
0.25 + x + x2 = 8.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 8.25

Calculate the square root of the right side: 2.872281323

Break this problem into two subproblems by setting 
(x + 0.5) equal to 2.872281323 and -2.872281323.

Subproblem 1

x + 0.5 = 2.872281323 Simplifying x + 0.5 = 2.872281323 Reorder the terms: 0.5 + x = 2.872281323 Solving 0.5 + x = 2.872281323 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 = 2.872281323 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 2.872281323 + -0.5 x = 2.872281323 + -0.5 Combine like terms: 2.872281323 + -0.5 = 2.372281323 x = 2.372281323 Simplifying x = 2.372281323

Subproblem 2

x + 0.5 = -2.872281323 Simplifying x + 0.5 = -2.872281323 Reorder the terms: 0.5 + x = -2.872281323 Solving 0.5 + x = -2.872281323 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 = -2.872281323 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -2.872281323 + -0.5 x = -2.872281323 + -0.5 Combine like terms: -2.872281323 + -0.5 = -3.372281323 x = -3.372281323 Simplifying x = -3.372281323

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.372281323, -3.372281323}

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