500=(6+2x)(x)

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Solution for 500=(6+2x)(x) equation: 


Simplifying
500 = (6 + 2x)(x)

Reorder the terms for easier multiplication:
500 = x(6 + 2x)
500 = (6 * x + 2x * x)
500 = (6x + 2x2)

Solving
500 = 6x + 2x2

Solving for variable 'x'.

Reorder the terms:
500 + -6x + -2x2 = 6x + -6x + 2x2 + -2x2

Combine like terms: 6x + -6x = 0
500 + -6x + -2x2 = 0 + 2x2 + -2x2
500 + -6x + -2x2 = 2x2 + -2x2

Combine like terms: 2x2 + -2x2 = 0
500 + -6x + -2x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(250 + -3x + -1x2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(250 + -3x + -1x2)' equal to zero and attempt to solve:

Simplifying
250 + -3x + -1x2 = 0

Solving
250 + -3x + -1x2 = 0

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

Divide each side by '-1'.
-250 + 3x + x2 = 0

Move the constant term to the right:

Add '250' to each side of the equation.
-250 + 3x + 250 + x2 = 0 + 250

Reorder the terms:
-250 + 250 + 3x + x2 = 0 + 250

Combine like terms: -250 + 250 = 0
0 + 3x + x2 = 0 + 250
3x + x2 = 0 + 250

Combine like terms: 0 + 250 = 250
3x + x2 = 250

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 = 250 + 2.25

Reorder the terms:
2.25 + 3x + x2 = 250 + 2.25

Combine like terms: 250 + 2.25 = 252.25
2.25 + 3x + x2 = 252.25

Factor a perfect square on the left side:
(x + 1.5)(x + 1.5) = 252.25

Calculate the square root of the right side: 15.882380174

Break this problem into two subproblems by setting 
(x + 1.5) equal to 15.882380174 and -15.882380174.


Subproblem 1
x + 1.5 = 15.882380174

Simplifying
x + 1.5 = 15.882380174

Reorder the terms:
1.5 + x = 15.882380174

Solving
1.5 + x = 15.882380174

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 = 15.882380174 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + x = 15.882380174 + -1.5
x = 15.882380174 + -1.5

Combine like terms: 15.882380174 + -1.5 = 14.382380174
x = 14.382380174

Simplifying
x = 14.382380174


Subproblem 2
x + 1.5 = -15.882380174

Simplifying
x + 1.5 = -15.882380174

Reorder the terms:
1.5 + x = -15.882380174

Solving
1.5 + x = -15.882380174

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 = -15.882380174 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + x = -15.882380174 + -1.5
x = -15.882380174 + -1.5

Combine like terms: -15.882380174 + -1.5 = -17.382380174
x = -17.382380174

Simplifying
x = -17.382380174


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {14.382380174, -17.382380174}
Solution
x = {14.382380174, -17.382380174}

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