100x(x+1)+50x=400

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Solution for 100x(x+1)+50x=400 equation: 


Simplifying
100x(x + 1) + 50x = 400

Reorder the terms:
100x(1 + x) + 50x = 400
(1 * 100x + x * 100x) + 50x = 400
(100x + 100x2) + 50x = 400

Reorder the terms:
100x + 50x + 100x2 = 400

Combine like terms: 100x + 50x = 150x
150x + 100x2 = 400

Solving
150x + 100x2 = 400

Solving for variable 'x'.

Reorder the terms:
-400 + 150x + 100x2 = 400 + -400

Combine like terms: 400 + -400 = 0
-400 + 150x + 100x2 = 0

Factor out the Greatest Common Factor (GCF), '50'.
50(-8 + 3x + 2x2) = 0

Ignore the factor 50.

Subproblem 1
Set the factor '(-8 + 3x + 2x2)' equal to zero and attempt to solve:

Simplifying
-8 + 3x + 2x2 = 0

Solving
-8 + 3x + 2x2 = 0

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

Divide each side by '2'.
-4 + 1.5x + x2 = 0

Move the constant term to the right:

Add '4' to each side of the equation.
-4 + 1.5x + 4 + x2 = 0 + 4

Reorder the terms:
-4 + 4 + 1.5x + x2 = 0 + 4

Combine like terms: -4 + 4 = 0
0 + 1.5x + x2 = 0 + 4
1.5x + x2 = 0 + 4

Combine like terms: 0 + 4 = 4
1.5x + x2 = 4

The x term is 1.5x.  Take half its coefficient (0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
1.5x + 0.5625 + x2 = 4 + 0.5625

Reorder the terms:
0.5625 + 1.5x + x2 = 4 + 0.5625

Combine like terms: 4 + 0.5625 = 4.5625
0.5625 + 1.5x + x2 = 4.5625

Factor a perfect square on the left side:
(x + 0.75)(x + 0.75) = 4.5625

Calculate the square root of the right side: 2.136000936

Break this problem into two subproblems by setting 
(x + 0.75) equal to 2.136000936 and -2.136000936.


Subproblem 1
x + 0.75 = 2.136000936

Simplifying
x + 0.75 = 2.136000936

Reorder the terms:
0.75 + x = 2.136000936

Solving
0.75 + x = 2.136000936

Solving for variable 'x'.

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

Add '-0.75' to each side of the equation.
0.75 + -0.75 + x = 2.136000936 + -0.75

Combine like terms: 0.75 + -0.75 = 0.00
0.00 + x = 2.136000936 + -0.75
x = 2.136000936 + -0.75

Combine like terms: 2.136000936 + -0.75 = 1.386000936
x = 1.386000936

Simplifying
x = 1.386000936


Subproblem 2
x + 0.75 = -2.136000936

Simplifying
x + 0.75 = -2.136000936

Reorder the terms:
0.75 + x = -2.136000936

Solving
0.75 + x = -2.136000936

Solving for variable 'x'.

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

Add '-0.75' to each side of the equation.
0.75 + -0.75 + x = -2.136000936 + -0.75

Combine like terms: 0.75 + -0.75 = 0.00
0.00 + x = -2.136000936 + -0.75
x = -2.136000936 + -0.75

Combine like terms: -2.136000936 + -0.75 = -2.886000936
x = -2.886000936

Simplifying
x = -2.886000936


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.386000936, -2.886000936}
Solution
x = {1.386000936, -2.886000936}

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