(6+2x)(4.5+2x)=137.5

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Solution for (6+2x)(4.5+2x)=137.5 equation: 


Simplifying
(6 + 2x)(4.5 + 2x) = 137.5

Multiply (6 + 2x) * (4.5 + 2x)
(6(4.5 + 2x) + 2x * (4.5 + 2x)) = 137.5
((4.5 * 6 + 2x * 6) + 2x * (4.5 + 2x)) = 137.5
((27 + 12x) + 2x * (4.5 + 2x)) = 137.5
(27 + 12x + (4.5 * 2x + 2x * 2x)) = 137.5
(27 + 12x + (9x + 4x2)) = 137.5

Combine like terms: 12x + 9x = 21x
(27 + 21x + 4x2) = 137.5

Solving
27 + 21x + 4x2 = 137.5

Solving for variable 'x'.

Reorder the terms:
27 + -137.5 + 21x + 4x2 = 137.5 + -137.5

Combine like terms: 27 + -137.5 = -110.5
-110.5 + 21x + 4x2 = 137.5 + -137.5

Combine like terms: 137.5 + -137.5 = 0.0
-110.5 + 21x + 4x2 = 0.0

Begin completing the square.  Divide all terms by
4 the coefficient of the squared term: 

Divide each side by '4'.
-27.625 + 5.25x + x2 = 0.0

Move the constant term to the right:

Add '27.625' to each side of the equation.
-27.625 + 5.25x + 27.625 + x2 = 0.0 + 27.625

Reorder the terms:
-27.625 + 27.625 + 5.25x + x2 = 0.0 + 27.625

Combine like terms: -27.625 + 27.625 = 0.000
0.000 + 5.25x + x2 = 0.0 + 27.625
5.25x + x2 = 0.0 + 27.625

Combine like terms: 0.0 + 27.625 = 27.625
5.25x + x2 = 27.625

The x term is 5.25x.  Take half its coefficient (2.625).
Square it (6.890625) and add it to both sides.

Add '6.890625' to each side of the equation.
5.25x + 6.890625 + x2 = 27.625 + 6.890625

Reorder the terms:
6.890625 + 5.25x + x2 = 27.625 + 6.890625

Combine like terms: 27.625 + 6.890625 = 34.515625
6.890625 + 5.25x + x2 = 34.515625

Factor a perfect square on the left side:
(x + 2.625)(x + 2.625) = 34.515625

Calculate the square root of the right side: 5.875

Break this problem into two subproblems by setting 
(x + 2.625) equal to 5.875 and -5.875.


Subproblem 1
x + 2.625 = 5.875

Simplifying
x + 2.625 = 5.875

Reorder the terms:
2.625 + x = 5.875

Solving
2.625 + x = 5.875

Solving for variable 'x'.

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

Add '-2.625' to each side of the equation.
2.625 + -2.625 + x = 5.875 + -2.625

Combine like terms: 2.625 + -2.625 = 0.000
0.000 + x = 5.875 + -2.625
x = 5.875 + -2.625

Combine like terms: 5.875 + -2.625 = 3.25
x = 3.25

Simplifying
x = 3.25


Subproblem 2
x + 2.625 = -5.875

Simplifying
x + 2.625 = -5.875

Reorder the terms:
2.625 + x = -5.875

Solving
2.625 + x = -5.875

Solving for variable 'x'.

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

Add '-2.625' to each side of the equation.
2.625 + -2.625 + x = -5.875 + -2.625

Combine like terms: 2.625 + -2.625 = 0.000
0.000 + x = -5.875 + -2.625
x = -5.875 + -2.625

Combine like terms: -5.875 + -2.625 = -8.5
x = -8.5

Simplifying
x = -8.5


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3.25, -8.5}

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