2(x+3)(x+5)-4=0

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Solution for 2(x+3)(x+5)-4=0 equation: 


Simplifying
2(x + 3)(x + 5) + -4 = 0

Reorder the terms:
2(3 + x)(x + 5) + -4 = 0

Reorder the terms:
2(3 + x)(5 + x) + -4 = 0

Multiply (3 + x) * (5 + x)
2(3(5 + x) + x(5 + x)) + -4 = 0
2((5 * 3 + x * 3) + x(5 + x)) + -4 = 0
2((15 + 3x) + x(5 + x)) + -4 = 0
2(15 + 3x + (5 * x + x * x)) + -4 = 0
2(15 + 3x + (5x + x2)) + -4 = 0

Combine like terms: 3x + 5x = 8x
2(15 + 8x + x2) + -4 = 0
(15 * 2 + 8x * 2 + x2 * 2) + -4 = 0
(30 + 16x + 2x2) + -4 = 0

Reorder the terms:
30 + -4 + 16x + 2x2 = 0

Combine like terms: 30 + -4 = 26
26 + 16x + 2x2 = 0

Solving
26 + 16x + 2x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '2'.
2(13 + 8x + x2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(13 + 8x + x2)' equal to zero and attempt to solve:

Simplifying
13 + 8x + x2 = 0

Solving
13 + 8x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-13' to each side of the equation.
13 + 8x + -13 + x2 = 0 + -13

Reorder the terms:
13 + -13 + 8x + x2 = 0 + -13

Combine like terms: 13 + -13 = 0
0 + 8x + x2 = 0 + -13
8x + x2 = 0 + -13

Combine like terms: 0 + -13 = -13
8x + x2 = -13

The x term is 8x.  Take half its coefficient (4).
Square it (16) and add it to both sides.

Add '16' to each side of the equation.
8x + 16 + x2 = -13 + 16

Reorder the terms:
16 + 8x + x2 = -13 + 16

Combine like terms: -13 + 16 = 3
16 + 8x + x2 = 3

Factor a perfect square on the left side:
(x + 4)(x + 4) = 3

Calculate the square root of the right side: 1.732050808

Break this problem into two subproblems by setting 
(x + 4) equal to 1.732050808 and -1.732050808.


Subproblem 1
x + 4 = 1.732050808

Simplifying
x + 4 = 1.732050808

Reorder the terms:
4 + x = 1.732050808

Solving
4 + x = 1.732050808

Solving for variable 'x'.

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

Add '-4' to each side of the equation.
4 + -4 + x = 1.732050808 + -4

Combine like terms: 4 + -4 = 0
0 + x = 1.732050808 + -4
x = 1.732050808 + -4

Combine like terms: 1.732050808 + -4 = -2.267949192
x = -2.267949192

Simplifying
x = -2.267949192


Subproblem 2
x + 4 = -1.732050808

Simplifying
x + 4 = -1.732050808

Reorder the terms:
4 + x = -1.732050808

Solving
4 + x = -1.732050808

Solving for variable 'x'.

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

Add '-4' to each side of the equation.
4 + -4 + x = -1.732050808 + -4

Combine like terms: 4 + -4 = 0
0 + x = -1.732050808 + -4
x = -1.732050808 + -4

Combine like terms: -1.732050808 + -4 = -5.732050808
x = -5.732050808

Simplifying
x = -5.732050808


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-2.267949192, -5.732050808}
Solution
x = {-2.267949192, -5.732050808}

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