x(2x+1)=4x-x+8

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Solution for x(2x+1)=4x-x+8 equation: 


Simplifying
x(2x + 1) = 4x + -1x + 8

Reorder the terms:
x(1 + 2x) = 4x + -1x + 8
(1 * x + 2x * x) = 4x + -1x + 8
(1x + 2x2) = 4x + -1x + 8

Reorder the terms:
1x + 2x2 = 8 + 4x + -1x

Combine like terms: 4x + -1x = 3x
1x + 2x2 = 8 + 3x

Solving
1x + 2x2 = 8 + 3x

Solving for variable 'x'.

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

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

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

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

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

Factor out the Greatest Common Factor (GCF), '2'.
2(-4 + -1x + x2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-4 + -1x + x2)' equal to zero and attempt to solve:

Simplifying
-4 + -1x + x2 = 0

Solving
-4 + -1x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '4' to each side of the equation.
-4 + -1x + 4 + x2 = 0 + 4

Reorder the terms:
-4 + 4 + -1x + x2 = 0 + 4

Combine like terms: -4 + 4 = 0
0 + -1x + x2 = 0 + 4
-1x + x2 = 0 + 4

Combine like terms: 0 + 4 = 4
-1x + x2 = 4

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.
-1x + 0.25 + x2 = 4 + 0.25

Reorder the terms:
0.25 + -1x + x2 = 4 + 0.25

Combine like terms: 4 + 0.25 = 4.25
0.25 + -1x + x2 = 4.25

Factor a perfect square on the left side:
(x + -0.5)(x + -0.5) = 4.25

Calculate the square root of the right side: 2.061552813

Break this problem into two subproblems by setting 
(x + -0.5) equal to 2.061552813 and -2.061552813.


Subproblem 1
x + -0.5 = 2.061552813

Simplifying
x + -0.5 = 2.061552813

Reorder the terms:
-0.5 + x = 2.061552813

Solving
-0.5 + x = 2.061552813

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 = 2.061552813 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + x = 2.061552813 + 0.5
x = 2.061552813 + 0.5

Combine like terms: 2.061552813 + 0.5 = 2.561552813
x = 2.561552813

Simplifying
x = 2.561552813


Subproblem 2
x + -0.5 = -2.061552813

Simplifying
x + -0.5 = -2.061552813

Reorder the terms:
-0.5 + x = -2.061552813

Solving
-0.5 + x = -2.061552813

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 = -2.061552813 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + x = -2.061552813 + 0.5
x = -2.061552813 + 0.5

Combine like terms: -2.061552813 + 0.5 = -1.561552813
x = -1.561552813

Simplifying
x = -1.561552813


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.561552813, -1.561552813}
Solution
x = {2.561552813, -1.561552813}

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