x^2=0.500(0.05+x)(0.1+x)

Solution for x^2=0.500(0.05+x)(0.1+x) equation:

Simplifying
x2 = 0.500(0.05 + x)(0.1 + x)

Multiply (0.05 + x) * (0.1 + x)
x2 = 0.500(0.05(0.1 + x) + x(0.1 + x))
x2 = 0.500((0.1 * 0.05 + x * 0.05) + x(0.1 + x))
x2 = 0.500((0.005 + 0.05x) + x(0.1 + x))
x2 = 0.500(0.005 + 0.05x + (0.1 * x + x * x))
x2 = 0.500(0.005 + 0.05x + (0.1x + x2))

Combine like terms: 0.05x + 0.1x = 0.15x
x2 = 0.500(0.005 + 0.15x + x2)
x2 = (0.005 * 0.500 + 0.15x * 0.500 + x2 * 0.500)
x2 = (0.0025 + 0.075x + 0.500x2)

Solving
x2 = 0.0025 + 0.075x + 0.500x2

Solving for variable 'x'.

Reorder the terms:
-0.0025 + -0.075x + x2 + -0.500x2 = 0.0025 + 0.075x + 0.500x2 + -0.0025 + -0.075x + -0.500x2

Combine like terms: x2 + -0.500x2 = 0.5x2
-0.0025 + -0.075x + 0.5x2 = 0.0025 + 0.075x + 0.500x2 + -0.0025 + -0.075x + -0.500x2

Reorder the terms:
-0.0025 + -0.075x + 0.5x2 = 0.0025 + -0.0025 + 0.075x + -0.075x + 0.500x2 + -0.500x2

Combine like terms: 0.0025 + -0.0025 = 0.0000
-0.0025 + -0.075x + 0.5x2 = 0.0000 + 0.075x + -0.075x + 0.500x2 + -0.500x2
-0.0025 + -0.075x + 0.5x2 = 0.075x + -0.075x + 0.500x2 + -0.500x2

Combine like terms: 0.075x + -0.075x = 0.000
-0.0025 + -0.075x + 0.5x2 = 0.000 + 0.500x2 + -0.500x2
-0.0025 + -0.075x + 0.5x2 = 0.500x2 + -0.500x2

Combine like terms: 0.500x2 + -0.500x2 = 0.000
-0.0025 + -0.075x + 0.5x2 = 0.000

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

Divide each side by '0.5'.
-0.005 + -0.15x + x2 = 0.00

Move the constant term to the right:

Add '0.005' to each side of the equation.
-0.005 + -0.15x + 0.005 + x2 = 0.00 + 0.005

Reorder the terms:
-0.005 + 0.005 + -0.15x + x2 = 0.00 + 0.005

Combine like terms: -0.005 + 0.005 = 0.000
0.000 + -0.15x + x2 = 0.00 + 0.005
-0.15x + x2 = 0.00 + 0.005

Combine like terms: 0.00 + 0.005 = 0.005
-0.15x + x2 = 0.005

The x term is -0.15x.  Take half its coefficient (-0.075).
Square it (0.005625) and add it to both sides.

Add '0.005625' to each side of the equation.
-0.15x + 0.005625 + x2 = 0.005 + 0.005625

Reorder the terms:
0.005625 + -0.15x + x2 = 0.005 + 0.005625

Combine like terms: 0.005 + 0.005625 = 0.010625
0.005625 + -0.15x + x2 = 0.010625

Factor a perfect square on the left side:
(x + -0.075)(x + -0.075) = 0.010625

Calculate the square root of the right side: 0.103077641

Break this problem into two subproblems by setting 
(x + -0.075) equal to 0.103077641 and -0.103077641.

Subproblem 1
x + -0.075 = 0.103077641

Simplifying
x + -0.075 = 0.103077641

Reorder the terms:
-0.075 + x = 0.103077641

Solving
-0.075 + x = 0.103077641

Solving for variable 'x'.

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

Add '0.075' to each side of the equation.
-0.075 + 0.075 + x = 0.103077641 + 0.075

Combine like terms: -0.075 + 0.075 = 0.000
0.000 + x = 0.103077641 + 0.075
x = 0.103077641 + 0.075

Combine like terms: 0.103077641 + 0.075 = 0.178077641
x = 0.178077641

Simplifying
x = 0.178077641

Subproblem 2
x + -0.075 = -0.103077641

Simplifying
x + -0.075 = -0.103077641

Reorder the terms:
-0.075 + x = -0.103077641

Solving
-0.075 + x = -0.103077641

Solving for variable 'x'.

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

Add '0.075' to each side of the equation.
-0.075 + 0.075 + x = -0.103077641 + 0.075

Combine like terms: -0.075 + 0.075 = 0.000
0.000 + x = -0.103077641 + 0.075
x = -0.103077641 + 0.075

Combine like terms: -0.103077641 + 0.075 = -0.028077641
x = -0.028077641

Simplifying
x = -0.028077641

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.178077641, -0.028077641}