(2x+3)(2x+4)=1

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Solution for (2x+3)(2x+4)=1 equation:


Simplifying
(2x + 3)(2x + 4) = 1

Reorder the terms:
(3 + 2x)(2x + 4) = 1

Reorder the terms:
(3 + 2x)(4 + 2x) = 1

Multiply (3 + 2x) * (4 + 2x)
(3(4 + 2x) + 2x * (4 + 2x)) = 1
((4 * 3 + 2x * 3) + 2x * (4 + 2x)) = 1
((12 + 6x) + 2x * (4 + 2x)) = 1
(12 + 6x + (4 * 2x + 2x * 2x)) = 1
(12 + 6x + (8x + 4x2)) = 1

Combine like terms: 6x + 8x = 14x
(12 + 14x + 4x2) = 1

Solving
12 + 14x + 4x2 = 1

Solving for variable 'x'.

Reorder the terms:
12 + -1 + 14x + 4x2 = 1 + -1

Combine like terms: 12 + -1 = 11
11 + 14x + 4x2 = 1 + -1

Combine like terms: 1 + -1 = 0
11 + 14x + 4x2 = 0

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

Divide each side by '4'.
2.75 + 3.5x + x2 = 0

Move the constant term to the right:

Add '-2.75' to each side of the equation.
2.75 + 3.5x + -2.75 + x2 = 0 + -2.75

Reorder the terms:
2.75 + -2.75 + 3.5x + x2 = 0 + -2.75

Combine like terms: 2.75 + -2.75 = 0.00
0.00 + 3.5x + x2 = 0 + -2.75
3.5x + x2 = 0 + -2.75

Combine like terms: 0 + -2.75 = -2.75
3.5x + x2 = -2.75

The x term is 3.5x.  Take half its coefficient (1.75).
Square it (3.0625) and add it to both sides.

Add '3.0625' to each side of the equation.
3.5x + 3.0625 + x2 = -2.75 + 3.0625

Reorder the terms:
3.0625 + 3.5x + x2 = -2.75 + 3.0625

Combine like terms: -2.75 + 3.0625 = 0.3125
3.0625 + 3.5x + x2 = 0.3125

Factor a perfect square on the left side:
(x + 1.75)(x + 1.75) = 0.3125

Calculate the square root of the right side: 0.559016994

Break this problem into two subproblems by setting 
(x + 1.75) equal to 0.559016994 and -0.559016994.

Subproblem 1

x + 1.75 = 0.559016994 Simplifying x + 1.75 = 0.559016994 Reorder the terms: 1.75 + x = 0.559016994 Solving 1.75 + x = 0.559016994 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + x = 0.559016994 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + x = 0.559016994 + -1.75 x = 0.559016994 + -1.75 Combine like terms: 0.559016994 + -1.75 = -1.190983006 x = -1.190983006 Simplifying x = -1.190983006

Subproblem 2

x + 1.75 = -0.559016994 Simplifying x + 1.75 = -0.559016994 Reorder the terms: 1.75 + x = -0.559016994 Solving 1.75 + x = -0.559016994 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + x = -0.559016994 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + x = -0.559016994 + -1.75 x = -0.559016994 + -1.75 Combine like terms: -0.559016994 + -1.75 = -2.309016994 x = -2.309016994 Simplifying x = -2.309016994

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-1.190983006, -2.309016994}

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