(4x+5)2x=100

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Solution for (4x+5)2x=100 equation: 


Simplifying
(4x + 5) * 2x = 100

Reorder the terms:
(5 + 4x) * 2x = 100

Reorder the terms for easier multiplication:
2x(5 + 4x) = 100
(5 * 2x + 4x * 2x) = 100
(10x + 8x2) = 100

Solving
10x + 8x2 = 100

Solving for variable 'x'.

Reorder the terms:
-100 + 10x + 8x2 = 100 + -100

Combine like terms: 100 + -100 = 0
-100 + 10x + 8x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-50 + 5x + 4x2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-50 + 5x + 4x2)' equal to zero and attempt to solve:

Simplifying
-50 + 5x + 4x2 = 0

Solving
-50 + 5x + 4x2 = 0

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

Divide each side by '4'.
-12.5 + 1.25x + x2 = 0

Move the constant term to the right:

Add '12.5' to each side of the equation.
-12.5 + 1.25x + 12.5 + x2 = 0 + 12.5

Reorder the terms:
-12.5 + 12.5 + 1.25x + x2 = 0 + 12.5

Combine like terms: -12.5 + 12.5 = 0.0
0.0 + 1.25x + x2 = 0 + 12.5
1.25x + x2 = 0 + 12.5

Combine like terms: 0 + 12.5 = 12.5
1.25x + x2 = 12.5

The x term is 1.25x.  Take half its coefficient (0.625).
Square it (0.390625) and add it to both sides.

Add '0.390625' to each side of the equation.
1.25x + 0.390625 + x2 = 12.5 + 0.390625

Reorder the terms:
0.390625 + 1.25x + x2 = 12.5 + 0.390625

Combine like terms: 12.5 + 0.390625 = 12.890625
0.390625 + 1.25x + x2 = 12.890625

Factor a perfect square on the left side:
(x + 0.625)(x + 0.625) = 12.890625

Calculate the square root of the right side: 3.590351654

Break this problem into two subproblems by setting 
(x + 0.625) equal to 3.590351654 and -3.590351654.


Subproblem 1
x + 0.625 = 3.590351654

Simplifying
x + 0.625 = 3.590351654

Reorder the terms:
0.625 + x = 3.590351654

Solving
0.625 + x = 3.590351654

Solving for variable 'x'.

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

Add '-0.625' to each side of the equation.
0.625 + -0.625 + x = 3.590351654 + -0.625

Combine like terms: 0.625 + -0.625 = 0.000
0.000 + x = 3.590351654 + -0.625
x = 3.590351654 + -0.625

Combine like terms: 3.590351654 + -0.625 = 2.965351654
x = 2.965351654

Simplifying
x = 2.965351654


Subproblem 2
x + 0.625 = -3.590351654

Simplifying
x + 0.625 = -3.590351654

Reorder the terms:
0.625 + x = -3.590351654

Solving
0.625 + x = -3.590351654

Solving for variable 'x'.

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

Add '-0.625' to each side of the equation.
0.625 + -0.625 + x = -3.590351654 + -0.625

Combine like terms: 0.625 + -0.625 = 0.000
0.000 + x = -3.590351654 + -0.625
x = -3.590351654 + -0.625

Combine like terms: -3.590351654 + -0.625 = -4.215351654
x = -4.215351654

Simplifying
x = -4.215351654


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.965351654, -4.215351654}
Solution
x = {2.965351654, -4.215351654}

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