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

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


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

Reorder the terms:
x(3 + 2x) = (x + 2)(x + 4)
(3 * x + 2x * x) = (x + 2)(x + 4)
(3x + 2x2) = (x + 2)(x + 4)

Reorder the terms:
3x + 2x2 = (2 + x)(x + 4)

Reorder the terms:
3x + 2x2 = (2 + x)(4 + x)

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

Combine like terms: 2x + 4x = 6x
3x + 2x2 = (8 + 6x + x2)

Solving
3x + 2x2 = 8 + 6x + x2

Solving for variable 'x'.

Reorder the terms:
-8 + 3x + -6x + 2x2 + -1x2 = 8 + 6x + x2 + -8 + -6x + -1x2

Combine like terms: 3x + -6x = -3x
-8 + -3x + 2x2 + -1x2 = 8 + 6x + x2 + -8 + -6x + -1x2

Combine like terms: 2x2 + -1x2 = 1x2
-8 + -3x + 1x2 = 8 + 6x + x2 + -8 + -6x + -1x2

Reorder the terms:
-8 + -3x + 1x2 = 8 + -8 + 6x + -6x + x2 + -1x2

Combine like terms: 8 + -8 = 0
-8 + -3x + 1x2 = 0 + 6x + -6x + x2 + -1x2
-8 + -3x + 1x2 = 6x + -6x + x2 + -1x2

Combine like terms: 6x + -6x = 0
-8 + -3x + 1x2 = 0 + x2 + -1x2
-8 + -3x + 1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
-8 + -3x + 1x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '8' to each side of the equation.
-8 + -3x + 8 + x2 = 0 + 8

Reorder the terms:
-8 + 8 + -3x + x2 = 0 + 8

Combine like terms: -8 + 8 = 0
0 + -3x + x2 = 0 + 8
-3x + x2 = 0 + 8

Combine like terms: 0 + 8 = 8
-3x + x2 = 8

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 = 8 + 2.25

Reorder the terms:
2.25 + -3x + x2 = 8 + 2.25

Combine like terms: 8 + 2.25 = 10.25
2.25 + -3x + x2 = 10.25

Factor a perfect square on the left side:
(x + -1.5)(x + -1.5) = 10.25

Calculate the square root of the right side: 3.201562119

Break this problem into two subproblems by setting 
(x + -1.5) equal to 3.201562119 and -3.201562119.

Subproblem 1

x + -1.5 = 3.201562119 Simplifying x + -1.5 = 3.201562119 Reorder the terms: -1.5 + x = 3.201562119 Solving -1.5 + x = 3.201562119 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 = 3.201562119 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = 3.201562119 + 1.5 x = 3.201562119 + 1.5 Combine like terms: 3.201562119 + 1.5 = 4.701562119 x = 4.701562119 Simplifying x = 4.701562119

Subproblem 2

x + -1.5 = -3.201562119 Simplifying x + -1.5 = -3.201562119 Reorder the terms: -1.5 + x = -3.201562119 Solving -1.5 + x = -3.201562119 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 = -3.201562119 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = -3.201562119 + 1.5 x = -3.201562119 + 1.5 Combine like terms: -3.201562119 + 1.5 = -1.701562119 x = -1.701562119 Simplifying x = -1.701562119

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4.701562119, -1.701562119}

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