100=4x(4x+1)

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Solution for 100=4x(4x+1) equation: 


Simplifying
100 = 4x(4x + 1)

Reorder the terms:
100 = 4x(1 + 4x)
100 = (1 * 4x + 4x * 4x)
100 = (4x + 16x2)

Solving
100 = 4x + 16x2

Solving for variable 'x'.

Reorder the terms:
100 + -4x + -16x2 = 4x + -4x + 16x2 + -16x2

Combine like terms: 4x + -4x = 0
100 + -4x + -16x2 = 0 + 16x2 + -16x2
100 + -4x + -16x2 = 16x2 + -16x2

Combine like terms: 16x2 + -16x2 = 0
100 + -4x + -16x2 = 0

Factor out the Greatest Common Factor (GCF), '4'.
4(25 + -1x + -4x2) = 0

Ignore the factor 4.

Subproblem 1
Set the factor '(25 + -1x + -4x2)' equal to zero and attempt to solve:

Simplifying
25 + -1x + -4x2 = 0

Solving
25 + -1x + -4x2 = 0

Begin completing the square.  Divide all terms by
-4 the coefficient of the squared term: 

Divide each side by '-4'.
-6.25 + 0.25x + x2 = 0

Move the constant term to the right:

Add '6.25' to each side of the equation.
-6.25 + 0.25x + 6.25 + x2 = 0 + 6.25

Reorder the terms:
-6.25 + 6.25 + 0.25x + x2 = 0 + 6.25

Combine like terms: -6.25 + 6.25 = 0.00
0.00 + 0.25x + x2 = 0 + 6.25
0.25x + x2 = 0 + 6.25

Combine like terms: 0 + 6.25 = 6.25
0.25x + x2 = 6.25

The x term is 0.25x.  Take half its coefficient (0.125).
Square it (0.015625) and add it to both sides.

Add '0.015625' to each side of the equation.
0.25x + 0.015625 + x2 = 6.25 + 0.015625

Reorder the terms:
0.015625 + 0.25x + x2 = 6.25 + 0.015625

Combine like terms: 6.25 + 0.015625 = 6.265625
0.015625 + 0.25x + x2 = 6.265625

Factor a perfect square on the left side:
(x + 0.125)(x + 0.125) = 6.265625

Calculate the square root of the right side: 2.503123049

Break this problem into two subproblems by setting 
(x + 0.125) equal to 2.503123049 and -2.503123049.


Subproblem 1
x + 0.125 = 2.503123049

Simplifying
x + 0.125 = 2.503123049

Reorder the terms:
0.125 + x = 2.503123049

Solving
0.125 + x = 2.503123049

Solving for variable 'x'.

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

Add '-0.125' to each side of the equation.
0.125 + -0.125 + x = 2.503123049 + -0.125

Combine like terms: 0.125 + -0.125 = 0.000
0.000 + x = 2.503123049 + -0.125
x = 2.503123049 + -0.125

Combine like terms: 2.503123049 + -0.125 = 2.378123049
x = 2.378123049

Simplifying
x = 2.378123049


Subproblem 2
x + 0.125 = -2.503123049

Simplifying
x + 0.125 = -2.503123049

Reorder the terms:
0.125 + x = -2.503123049

Solving
0.125 + x = -2.503123049

Solving for variable 'x'.

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

Add '-0.125' to each side of the equation.
0.125 + -0.125 + x = -2.503123049 + -0.125

Combine like terms: 0.125 + -0.125 = 0.000
0.000 + x = -2.503123049 + -0.125
x = -2.503123049 + -0.125

Combine like terms: -2.503123049 + -0.125 = -2.628123049
x = -2.628123049

Simplifying
x = -2.628123049


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.378123049, -2.628123049}
Solution
x = {2.378123049, -2.628123049}

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