400=2x(5+x)

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Solution for 400=2x(5+x) equation: 


Simplifying
400 = 2x(5 + x)
400 = (5 * 2x + x * 2x)
400 = (10x + 2x2)

Solving
400 = 10x + 2x2

Solving for variable 'x'.

Reorder the terms:
400 + -10x + -2x2 = 10x + -10x + 2x2 + -2x2

Combine like terms: 10x + -10x = 0
400 + -10x + -2x2 = 0 + 2x2 + -2x2
400 + -10x + -2x2 = 2x2 + -2x2

Combine like terms: 2x2 + -2x2 = 0
400 + -10x + -2x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(200 + -5x + -1x2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(200 + -5x + -1x2)' equal to zero and attempt to solve:

Simplifying
200 + -5x + -1x2 = 0

Solving
200 + -5x + -1x2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-200 + 5x + x2 = 0

Move the constant term to the right:

Add '200' to each side of the equation.
-200 + 5x + 200 + x2 = 0 + 200

Reorder the terms:
-200 + 200 + 5x + x2 = 0 + 200

Combine like terms: -200 + 200 = 0
0 + 5x + x2 = 0 + 200
5x + x2 = 0 + 200

Combine like terms: 0 + 200 = 200
5x + x2 = 200

The x term is 5x.  Take half its coefficient (2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
5x + 6.25 + x2 = 200 + 6.25

Reorder the terms:
6.25 + 5x + x2 = 200 + 6.25

Combine like terms: 200 + 6.25 = 206.25
6.25 + 5x + x2 = 206.25

Factor a perfect square on the left side:
(x + 2.5)(x + 2.5) = 206.25

Calculate the square root of the right side: 14.361406616

Break this problem into two subproblems by setting 
(x + 2.5) equal to 14.361406616 and -14.361406616.


Subproblem 1
x + 2.5 = 14.361406616

Simplifying
x + 2.5 = 14.361406616

Reorder the terms:
2.5 + x = 14.361406616

Solving
2.5 + x = 14.361406616

Solving for variable 'x'.

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

Add '-2.5' to each side of the equation.
2.5 + -2.5 + x = 14.361406616 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + x = 14.361406616 + -2.5
x = 14.361406616 + -2.5

Combine like terms: 14.361406616 + -2.5 = 11.861406616
x = 11.861406616

Simplifying
x = 11.861406616


Subproblem 2
x + 2.5 = -14.361406616

Simplifying
x + 2.5 = -14.361406616

Reorder the terms:
2.5 + x = -14.361406616

Solving
2.5 + x = -14.361406616

Solving for variable 'x'.

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

Add '-2.5' to each side of the equation.
2.5 + -2.5 + x = -14.361406616 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + x = -14.361406616 + -2.5
x = -14.361406616 + -2.5

Combine like terms: -14.361406616 + -2.5 = -16.861406616
x = -16.861406616

Simplifying
x = -16.861406616


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {11.861406616, -16.861406616}
Solution
x = {11.861406616, -16.861406616}

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