-x^2=2x+(3x+2)

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Solution for -x^2=2x+(3x+2) equation: 


Simplifying
-1x2 = 2x + (3x + 2)

Reorder the terms:
-1x2 = 2x + (2 + 3x)

Remove parenthesis around (2 + 3x)
-1x2 = 2x + 2 + 3x

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

Combine like terms: 2x + 3x = 5x
-1x2 = 2 + 5x

Solving
-1x2 = 2 + 5x

Solving for variable 'x'.

Reorder the terms:
-2 + -5x + -1x2 = 2 + 5x + -2 + -5x

Reorder the terms:
-2 + -5x + -1x2 = 2 + -2 + 5x + -5x

Combine like terms: 2 + -2 = 0
-2 + -5x + -1x2 = 0 + 5x + -5x
-2 + -5x + -1x2 = 5x + -5x

Combine like terms: 5x + -5x = 0
-2 + -5x + -1x2 = 0

Factor out the Greatest Common Factor (GCF), '-1'.
-1(2 + 5x + x2) = 0

Ignore the factor -1.

Subproblem 1
Set the factor '(2 + 5x + x2)' equal to zero and attempt to solve:

Simplifying
2 + 5x + x2 = 0

Solving
2 + 5x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-2' to each side of the equation.
2 + 5x + -2 + x2 = 0 + -2

Reorder the terms:
2 + -2 + 5x + x2 = 0 + -2

Combine like terms: 2 + -2 = 0
0 + 5x + x2 = 0 + -2
5x + x2 = 0 + -2

Combine like terms: 0 + -2 = -2
5x + x2 = -2

The x term is 5x.  Take half its coefficient (2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
5x + 6.25 + x2 = -2 + 6.25

Reorder the terms:
6.25 + 5x + x2 = -2 + 6.25

Combine like terms: -2 + 6.25 = 4.25
6.25 + 5x + x2 = 4.25

Factor a perfect square on the left side:
(x + 2.5)(x + 2.5) = 4.25

Calculate the square root of the right side: 2.061552813

Break this problem into two subproblems by setting 
(x + 2.5) equal to 2.061552813 and -2.061552813.


Subproblem 1
x + 2.5 = 2.061552813

Simplifying
x + 2.5 = 2.061552813

Reorder the terms:
2.5 + x = 2.061552813

Solving
2.5 + x = 2.061552813

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-2.5' to each side of the equation.
2.5 + -2.5 + x = 2.061552813 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + x = 2.061552813 + -2.5
x = 2.061552813 + -2.5

Combine like terms: 2.061552813 + -2.5 = -0.438447187
x = -0.438447187

Simplifying
x = -0.438447187


Subproblem 2
x + 2.5 = -2.061552813

Simplifying
x + 2.5 = -2.061552813

Reorder the terms:
2.5 + x = -2.061552813

Solving
2.5 + x = -2.061552813

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-2.5' to each side of the equation.
2.5 + -2.5 + x = -2.061552813 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + x = -2.061552813 + -2.5
x = -2.061552813 + -2.5

Combine like terms: -2.061552813 + -2.5 = -4.561552813
x = -4.561552813

Simplifying
x = -4.561552813


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.438447187, -4.561552813}
Solution
x = {-0.438447187, -4.561552813}

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