(20+2x)(24+2x)=416

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Solution for (20+2x)(24+2x)=416 equation: 


Simplifying
(20 + 2x)(24 + 2x) = 416

Multiply (20 + 2x) * (24 + 2x)
(20(24 + 2x) + 2x * (24 + 2x)) = 416
((24 * 20 + 2x * 20) + 2x * (24 + 2x)) = 416
((480 + 40x) + 2x * (24 + 2x)) = 416
(480 + 40x + (24 * 2x + 2x * 2x)) = 416
(480 + 40x + (48x + 4x2)) = 416

Combine like terms: 40x + 48x = 88x
(480 + 88x + 4x2) = 416

Solving
480 + 88x + 4x2 = 416

Solving for variable 'x'.

Reorder the terms:
480 + -416 + 88x + 4x2 = 416 + -416

Combine like terms: 480 + -416 = 64
64 + 88x + 4x2 = 416 + -416

Combine like terms: 416 + -416 = 0
64 + 88x + 4x2 = 0

Factor out the Greatest Common Factor (GCF), '4'.
4(16 + 22x + x2) = 0

Ignore the factor 4.

Subproblem 1
Set the factor '(16 + 22x + x2)' equal to zero and attempt to solve:

Simplifying
16 + 22x + x2 = 0

Solving
16 + 22x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-16' to each side of the equation.
16 + 22x + -16 + x2 = 0 + -16

Reorder the terms:
16 + -16 + 22x + x2 = 0 + -16

Combine like terms: 16 + -16 = 0
0 + 22x + x2 = 0 + -16
22x + x2 = 0 + -16

Combine like terms: 0 + -16 = -16
22x + x2 = -16

The x term is 22x.  Take half its coefficient (11).
Square it (121) and add it to both sides.

Add '121' to each side of the equation.
22x + 121 + x2 = -16 + 121

Reorder the terms:
121 + 22x + x2 = -16 + 121

Combine like terms: -16 + 121 = 105
121 + 22x + x2 = 105

Factor a perfect square on the left side:
(x + 11)(x + 11) = 105

Calculate the square root of the right side: 10.246950766

Break this problem into two subproblems by setting 
(x + 11) equal to 10.246950766 and -10.246950766.


Subproblem 1
x + 11 = 10.246950766

Simplifying
x + 11 = 10.246950766

Reorder the terms:
11 + x = 10.246950766

Solving
11 + x = 10.246950766

Solving for variable 'x'.

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

Add '-11' to each side of the equation.
11 + -11 + x = 10.246950766 + -11

Combine like terms: 11 + -11 = 0
0 + x = 10.246950766 + -11
x = 10.246950766 + -11

Combine like terms: 10.246950766 + -11 = -0.753049234
x = -0.753049234

Simplifying
x = -0.753049234


Subproblem 2
x + 11 = -10.246950766

Simplifying
x + 11 = -10.246950766

Reorder the terms:
11 + x = -10.246950766

Solving
11 + x = -10.246950766

Solving for variable 'x'.

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

Add '-11' to each side of the equation.
11 + -11 + x = -10.246950766 + -11

Combine like terms: 11 + -11 = 0
0 + x = -10.246950766 + -11
x = -10.246950766 + -11

Combine like terms: -10.246950766 + -11 = -21.246950766
x = -21.246950766

Simplifying
x = -21.246950766


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.753049234, -21.246950766}
Solution
x = {-0.753049234, -21.246950766}

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