(5x+1)4x=1000

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Solution for (5x+1)4x=1000 equation: 


Simplifying
(5x + 1) * 4x = 1000

Reorder the terms:
(1 + 5x) * 4x = 1000

Reorder the terms for easier multiplication:
4x(1 + 5x) = 1000
(1 * 4x + 5x * 4x) = 1000
(4x + 20x2) = 1000

Solving
4x + 20x2 = 1000

Solving for variable 'x'.

Reorder the terms:
-1000 + 4x + 20x2 = 1000 + -1000

Combine like terms: 1000 + -1000 = 0
-1000 + 4x + 20x2 = 0

Factor out the Greatest Common Factor (GCF), '4'.
4(-250 + x + 5x2) = 0

Ignore the factor 4.

Subproblem 1
Set the factor '(-250 + x + 5x2)' equal to zero and attempt to solve:

Simplifying
-250 + x + 5x2 = 0

Solving
-250 + x + 5x2 = 0

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

Divide each side by '5'.
-50 + 0.2x + x2 = 0

Move the constant term to the right:

Add '50' to each side of the equation.
-50 + 0.2x + 50 + x2 = 0 + 50

Reorder the terms:
-50 + 50 + 0.2x + x2 = 0 + 50

Combine like terms: -50 + 50 = 0
0 + 0.2x + x2 = 0 + 50
0.2x + x2 = 0 + 50

Combine like terms: 0 + 50 = 50
0.2x + x2 = 50

The x term is x.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
0.2x + 0.25 + x2 = 50 + 0.25

Reorder the terms:
0.25 + 0.2x + x2 = 50 + 0.25

Combine like terms: 50 + 0.25 = 50.25
0.25 + 0.2x + x2 = 50.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 50.25

Calculate the square root of the right side: 7.088723439

Break this problem into two subproblems by setting 
(x + 0.5) equal to 7.088723439 and -7.088723439.


Subproblem 1
x + 0.5 = 7.088723439

Simplifying
x + 0.5 = 7.088723439

Reorder the terms:
0.5 + x = 7.088723439

Solving
0.5 + x = 7.088723439

Solving for variable 'x'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = 7.088723439 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 7.088723439 + -0.5
x = 7.088723439 + -0.5

Combine like terms: 7.088723439 + -0.5 = 6.588723439
x = 6.588723439

Simplifying
x = 6.588723439


Subproblem 2
x + 0.5 = -7.088723439

Simplifying
x + 0.5 = -7.088723439

Reorder the terms:
0.5 + x = -7.088723439

Solving
0.5 + x = -7.088723439

Solving for variable 'x'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = -7.088723439 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -7.088723439 + -0.5
x = -7.088723439 + -0.5

Combine like terms: -7.088723439 + -0.5 = -7.588723439
x = -7.588723439

Simplifying
x = -7.588723439


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {6.588723439, -7.588723439}
Solution
x = {6.588723439, -7.588723439}

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