x^2+4x-6=2x+10-x^2

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Solution for x^2+4x-6=2x+10-x^2 equation: 


Simplifying
x2 + 4x + -6 = 2x + 10 + -1x2

Reorder the terms:
-6 + 4x + x2 = 2x + 10 + -1x2

Reorder the terms:
-6 + 4x + x2 = 10 + 2x + -1x2

Solving
-6 + 4x + x2 = 10 + 2x + -1x2

Solving for variable 'x'.

Reorder the terms:
-6 + -10 + 4x + -2x + x2 + x2 = 10 + 2x + -1x2 + -10 + -2x + x2

Combine like terms: -6 + -10 = -16
-16 + 4x + -2x + x2 + x2 = 10 + 2x + -1x2 + -10 + -2x + x2

Combine like terms: 4x + -2x = 2x
-16 + 2x + x2 + x2 = 10 + 2x + -1x2 + -10 + -2x + x2

Combine like terms: x2 + x2 = 2x2
-16 + 2x + 2x2 = 10 + 2x + -1x2 + -10 + -2x + x2

Reorder the terms:
-16 + 2x + 2x2 = 10 + -10 + 2x + -2x + -1x2 + x2

Combine like terms: 10 + -10 = 0
-16 + 2x + 2x2 = 0 + 2x + -2x + -1x2 + x2
-16 + 2x + 2x2 = 2x + -2x + -1x2 + x2

Combine like terms: 2x + -2x = 0
-16 + 2x + 2x2 = 0 + -1x2 + x2
-16 + 2x + 2x2 = -1x2 + x2

Combine like terms: -1x2 + x2 = 0
-16 + 2x + 2x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-8 + x + x2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-8 + x + x2)' equal to zero and attempt to solve:

Simplifying
-8 + x + x2 = 0

Solving
-8 + x + x2 = 0

Begin completing the square.

Move the constant term to the right:

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

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

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

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

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 = 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}
Solution
x = {2.372281323, -3.372281323}

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