4x(5x+1)=500

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Solution for 4x(5x+1)=500 equation: 


Simplifying
4x(5x + 1) = 500

Reorder the terms:
4x(1 + 5x) = 500
(1 * 4x + 5x * 4x) = 500
(4x + 20x2) = 500

Solving
4x + 20x2 = 500

Solving for variable 'x'.

Reorder the terms:
-500 + 4x + 20x2 = 500 + -500

Combine like terms: 500 + -500 = 0
-500 + 4x + 20x2 = 0

Factor out the Greatest Common Factor (GCF), '4'.
4(-125 + x + 5x2) = 0

Ignore the factor 4.

Subproblem 1
Set the factor '(-125 + x + 5x2)' equal to zero and attempt to solve:

Simplifying
-125 + x + 5x2 = 0

Solving
-125 + x + 5x2 = 0

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

Divide each side by '5'.
-25 + 0.2x + x2 = 0

Move the constant term to the right:

Add '25' to each side of the equation.
-25 + 0.2x + 25 + x2 = 0 + 25

Reorder the terms:
-25 + 25 + 0.2x + x2 = 0 + 25

Combine like terms: -25 + 25 = 0
0 + 0.2x + x2 = 0 + 25
0.2x + x2 = 0 + 25

Combine like terms: 0 + 25 = 25
0.2x + x2 = 25

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 = 25 + 0.25

Reorder the terms:
0.25 + 0.2x + x2 = 25 + 0.25

Combine like terms: 25 + 0.25 = 25.25
0.25 + 0.2x + x2 = 25.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 25.25

Calculate the square root of the right side: 5.024937811

Break this problem into two subproblems by setting 
(x + 0.5) equal to 5.024937811 and -5.024937811.


Subproblem 1
x + 0.5 = 5.024937811

Simplifying
x + 0.5 = 5.024937811

Reorder the terms:
0.5 + x = 5.024937811

Solving
0.5 + x = 5.024937811

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 = 5.024937811 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 5.024937811 + -0.5
x = 5.024937811 + -0.5

Combine like terms: 5.024937811 + -0.5 = 4.524937811
x = 4.524937811

Simplifying
x = 4.524937811


Subproblem 2
x + 0.5 = -5.024937811

Simplifying
x + 0.5 = -5.024937811

Reorder the terms:
0.5 + x = -5.024937811

Solving
0.5 + x = -5.024937811

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 = -5.024937811 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -5.024937811 + -0.5
x = -5.024937811 + -0.5

Combine like terms: -5.024937811 + -0.5 = -5.524937811
x = -5.524937811

Simplifying
x = -5.524937811


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {4.524937811, -5.524937811}
Solution
x = {4.524937811, -5.524937811}

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