(2x+3)(2x+6)=324

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Solution for (2x+3)(2x+6)=324 equation: 


Simplifying
(2x + 3)(2x + 6) = 324

Reorder the terms:
(3 + 2x)(2x + 6) = 324

Reorder the terms:
(3 + 2x)(6 + 2x) = 324

Multiply (3 + 2x) * (6 + 2x)
(3(6 + 2x) + 2x * (6 + 2x)) = 324
((6 * 3 + 2x * 3) + 2x * (6 + 2x)) = 324
((18 + 6x) + 2x * (6 + 2x)) = 324
(18 + 6x + (6 * 2x + 2x * 2x)) = 324
(18 + 6x + (12x + 4x2)) = 324

Combine like terms: 6x + 12x = 18x
(18 + 18x + 4x2) = 324

Solving
18 + 18x + 4x2 = 324

Solving for variable 'x'.

Reorder the terms:
18 + -324 + 18x + 4x2 = 324 + -324

Combine like terms: 18 + -324 = -306
-306 + 18x + 4x2 = 324 + -324

Combine like terms: 324 + -324 = 0
-306 + 18x + 4x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-153 + 9x + 2x2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-153 + 9x + 2x2)' equal to zero and attempt to solve:

Simplifying
-153 + 9x + 2x2 = 0

Solving
-153 + 9x + 2x2 = 0

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

Divide each side by '2'.
-76.5 + 4.5x + x2 = 0

Move the constant term to the right:

Add '76.5' to each side of the equation.
-76.5 + 4.5x + 76.5 + x2 = 0 + 76.5

Reorder the terms:
-76.5 + 76.5 + 4.5x + x2 = 0 + 76.5

Combine like terms: -76.5 + 76.5 = 0.0
0.0 + 4.5x + x2 = 0 + 76.5
4.5x + x2 = 0 + 76.5

Combine like terms: 0 + 76.5 = 76.5
4.5x + x2 = 76.5

The x term is 4.5x.  Take half its coefficient (2.25).
Square it (5.0625) and add it to both sides.

Add '5.0625' to each side of the equation.
4.5x + 5.0625 + x2 = 76.5 + 5.0625

Reorder the terms:
5.0625 + 4.5x + x2 = 76.5 + 5.0625

Combine like terms: 76.5 + 5.0625 = 81.5625
5.0625 + 4.5x + x2 = 81.5625

Factor a perfect square on the left side:
(x + 2.25)(x + 2.25) = 81.5625

Calculate the square root of the right side: 9.031195934

Break this problem into two subproblems by setting 
(x + 2.25) equal to 9.031195934 and -9.031195934.


Subproblem 1
x + 2.25 = 9.031195934

Simplifying
x + 2.25 = 9.031195934

Reorder the terms:
2.25 + x = 9.031195934

Solving
2.25 + x = 9.031195934

Solving for variable 'x'.

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

Add '-2.25' to each side of the equation.
2.25 + -2.25 + x = 9.031195934 + -2.25

Combine like terms: 2.25 + -2.25 = 0.00
0.00 + x = 9.031195934 + -2.25
x = 9.031195934 + -2.25

Combine like terms: 9.031195934 + -2.25 = 6.781195934
x = 6.781195934

Simplifying
x = 6.781195934


Subproblem 2
x + 2.25 = -9.031195934

Simplifying
x + 2.25 = -9.031195934

Reorder the terms:
2.25 + x = -9.031195934

Solving
2.25 + x = -9.031195934

Solving for variable 'x'.

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

Add '-2.25' to each side of the equation.
2.25 + -2.25 + x = -9.031195934 + -2.25

Combine like terms: 2.25 + -2.25 = 0.00
0.00 + x = -9.031195934 + -2.25
x = -9.031195934 + -2.25

Combine like terms: -9.031195934 + -2.25 = -11.281195934
x = -11.281195934

Simplifying
x = -11.281195934


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {6.781195934, -11.281195934}
Solution
x = {6.781195934, -11.281195934}

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