(x+2)(x-4)+3x+1=0

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Solution for (x+2)(x-4)+3x+1=0 equation: 


Simplifying
(x + 2)(x + -4) + 3x + 1 = 0

Reorder the terms:
(2 + x)(x + -4) + 3x + 1 = 0

Reorder the terms:
(2 + x)(-4 + x) + 3x + 1 = 0

Multiply (2 + x) * (-4 + x)
(2(-4 + x) + x(-4 + x)) + 3x + 1 = 0
((-4 * 2 + x * 2) + x(-4 + x)) + 3x + 1 = 0
((-8 + 2x) + x(-4 + x)) + 3x + 1 = 0
(-8 + 2x + (-4 * x + x * x)) + 3x + 1 = 0
(-8 + 2x + (-4x + x2)) + 3x + 1 = 0

Combine like terms: 2x + -4x = -2x
(-8 + -2x + x2) + 3x + 1 = 0

Reorder the terms:
-8 + 1 + -2x + 3x + x2 = 0

Combine like terms: -8 + 1 = -7
-7 + -2x + 3x + x2 = 0

Combine like terms: -2x + 3x = 1x
-7 + 1x + x2 = 0

Solving
-7 + 1x + x2 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '7' to each side of the equation.
-7 + 1x + 7 + x2 = 0 + 7

Reorder the terms:
-7 + 7 + 1x + x2 = 0 + 7

Combine like terms: -7 + 7 = 0
0 + 1x + x2 = 0 + 7
1x + x2 = 0 + 7

Combine like terms: 0 + 7 = 7
1x + x2 = 7

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 = 7 + 0.25

Reorder the terms:
0.25 + x + x2 = 7 + 0.25

Combine like terms: 7 + 0.25 = 7.25
0.25 + x + x2 = 7.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 7.25

Calculate the square root of the right side: 2.692582404

Break this problem into two subproblems by setting 
(x + 0.5) equal to 2.692582404 and -2.692582404.


Subproblem 1
x + 0.5 = 2.692582404

Simplifying
x + 0.5 = 2.692582404

Reorder the terms:
0.5 + x = 2.692582404

Solving
0.5 + x = 2.692582404

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 = 2.692582404 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 2.692582404 + -0.5
x = 2.692582404 + -0.5

Combine like terms: 2.692582404 + -0.5 = 2.192582404
x = 2.192582404

Simplifying
x = 2.192582404


Subproblem 2
x + 0.5 = -2.692582404

Simplifying
x + 0.5 = -2.692582404

Reorder the terms:
0.5 + x = -2.692582404

Solving
0.5 + x = -2.692582404

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 = -2.692582404 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -2.692582404 + -0.5
x = -2.692582404 + -0.5

Combine like terms: -2.692582404 + -0.5 = -3.192582404
x = -3.192582404

Simplifying
x = -3.192582404


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.192582404, -3.192582404}

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