(2x+3)(2x+3)=5

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


Simplifying
(2x + 3)(2x + 3) = 5

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

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

Multiply (3 + 2x) * (3 + 2x)
(3(3 + 2x) + 2x * (3 + 2x)) = 5
((3 * 3 + 2x * 3) + 2x * (3 + 2x)) = 5
((9 + 6x) + 2x * (3 + 2x)) = 5
(9 + 6x + (3 * 2x + 2x * 2x)) = 5
(9 + 6x + (6x + 4x2)) = 5

Combine like terms: 6x + 6x = 12x
(9 + 12x + 4x2) = 5

Solving
9 + 12x + 4x2 = 5

Solving for variable 'x'.

Reorder the terms:
9 + -5 + 12x + 4x2 = 5 + -5

Combine like terms: 9 + -5 = 4
4 + 12x + 4x2 = 5 + -5

Combine like terms: 5 + -5 = 0
4 + 12x + 4x2 = 0

Factor out the Greatest Common Factor (GCF), '4'.
4(1 + 3x + x2) = 0

Ignore the factor 4.

Subproblem 1
Set the factor '(1 + 3x + x2)' equal to zero and attempt to solve:

Simplifying
1 + 3x + x2 = 0

Solving
1 + 3x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + 3x + -1 + x2 = 0 + -1

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

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

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

The x term is 3x.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3x + 2.25 + x2 = -1 + 2.25

Reorder the terms:
2.25 + 3x + x2 = -1 + 2.25

Combine like terms: -1 + 2.25 = 1.25
2.25 + 3x + x2 = 1.25

Factor a perfect square on the left side:
(x + 1.5)(x + 1.5) = 1.25

Calculate the square root of the right side: 1.118033989

Break this problem into two subproblems by setting 
(x + 1.5) equal to 1.118033989 and -1.118033989.


Subproblem 1
x + 1.5 = 1.118033989

Simplifying
x + 1.5 = 1.118033989

Reorder the terms:
1.5 + x = 1.118033989

Solving
1.5 + x = 1.118033989

Solving for variable 'x'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + x = 1.118033989 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + x = 1.118033989 + -1.5
x = 1.118033989 + -1.5

Combine like terms: 1.118033989 + -1.5 = -0.381966011
x = -0.381966011

Simplifying
x = -0.381966011


Subproblem 2
x + 1.5 = -1.118033989

Simplifying
x + 1.5 = -1.118033989

Reorder the terms:
1.5 + x = -1.118033989

Solving
1.5 + x = -1.118033989

Solving for variable 'x'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + x = -1.118033989 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + x = -1.118033989 + -1.5
x = -1.118033989 + -1.5

Combine like terms: -1.118033989 + -1.5 = -2.618033989
x = -2.618033989

Simplifying
x = -2.618033989


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.381966011, -2.618033989}
Solution
x = {-0.381966011, -2.618033989}

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