2((2+x)*(3+x))=16+x

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Solution for 2((2+x)*(3+x))=16+x equation:


Simplifying
2((2 + x)(3 + x)) = 16 + x

Multiply (2 + x) * (3 + x)
2((2(3 + x) + x(3 + x))) = 16 + x
2(((3 * 2 + x * 2) + x(3 + x))) = 16 + x
2(((6 + 2x) + x(3 + x))) = 16 + x
2((6 + 2x + (3 * x + x * x))) = 16 + x
2((6 + 2x + (3x + x2))) = 16 + x

Combine like terms: 2x + 3x = 5x
2((6 + 5x + x2)) = 16 + x
(6 * 2 + 5x * 2 + x2 * 2) = 16 + x
(12 + 10x + 2x2) = 16 + x

Solving
12 + 10x + 2x2 = 16 + x

Solving for variable 'x'.

Reorder the terms:
12 + -16 + 10x + -1x + 2x2 = 16 + x + -16 + -1x

Combine like terms: 12 + -16 = -4
-4 + 10x + -1x + 2x2 = 16 + x + -16 + -1x

Combine like terms: 10x + -1x = 9x
-4 + 9x + 2x2 = 16 + x + -16 + -1x

Reorder the terms:
-4 + 9x + 2x2 = 16 + -16 + x + -1x

Combine like terms: 16 + -16 = 0
-4 + 9x + 2x2 = 0 + x + -1x
-4 + 9x + 2x2 = x + -1x

Combine like terms: x + -1x = 0
-4 + 9x + 2x2 = 0

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

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

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + 4.5x + 2 + x2 = 0 + 2

Reorder the terms:
-2 + 2 + 4.5x + x2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + 4.5x + x2 = 0 + 2
4.5x + x2 = 0 + 2

Combine like terms: 0 + 2 = 2
4.5x + x2 = 2

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 = 2 + 5.0625

Reorder the terms:
5.0625 + 4.5x + x2 = 2 + 5.0625

Combine like terms: 2 + 5.0625 = 7.0625
5.0625 + 4.5x + x2 = 7.0625

Factor a perfect square on the left side:
(x + 2.25)(x + 2.25) = 7.0625

Calculate the square root of the right side: 2.657536453

Break this problem into two subproblems by setting 
(x + 2.25) equal to 2.657536453 and -2.657536453.

Subproblem 1

x + 2.25 = 2.657536453 Simplifying x + 2.25 = 2.657536453 Reorder the terms: 2.25 + x = 2.657536453 Solving 2.25 + x = 2.657536453 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 = 2.657536453 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + x = 2.657536453 + -2.25 x = 2.657536453 + -2.25 Combine like terms: 2.657536453 + -2.25 = 0.407536453 x = 0.407536453 Simplifying x = 0.407536453

Subproblem 2

x + 2.25 = -2.657536453 Simplifying x + 2.25 = -2.657536453 Reorder the terms: 2.25 + x = -2.657536453 Solving 2.25 + x = -2.657536453 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 = -2.657536453 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + x = -2.657536453 + -2.25 x = -2.657536453 + -2.25 Combine like terms: -2.657536453 + -2.25 = -4.907536453 x = -4.907536453 Simplifying x = -4.907536453

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.407536453, -4.907536453}

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