4x(2x-3)=5

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Solution for 4x(2x-3)=5 equation: 


Simplifying
4x(2x + -3) = 5

Reorder the terms:
4x(-3 + 2x) = 5
(-3 * 4x + 2x * 4x) = 5
(-12x + 8x2) = 5

Solving
-12x + 8x2 = 5

Solving for variable 'x'.

Reorder the terms:
-5 + -12x + 8x2 = 5 + -5

Combine like terms: 5 + -5 = 0
-5 + -12x + 8x2 = 0

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

Divide each side by '8'.
-0.625 + -1.5x + x2 = 0

Move the constant term to the right:

Add '0.625' to each side of the equation.
-0.625 + -1.5x + 0.625 + x2 = 0 + 0.625

Reorder the terms:
-0.625 + 0.625 + -1.5x + x2 = 0 + 0.625

Combine like terms: -0.625 + 0.625 = 0.000
0.000 + -1.5x + x2 = 0 + 0.625
-1.5x + x2 = 0 + 0.625

Combine like terms: 0 + 0.625 = 0.625
-1.5x + x2 = 0.625

The x term is -1.5x.  Take half its coefficient (-0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
-1.5x + 0.5625 + x2 = 0.625 + 0.5625

Reorder the terms:
0.5625 + -1.5x + x2 = 0.625 + 0.5625

Combine like terms: 0.625 + 0.5625 = 1.1875
0.5625 + -1.5x + x2 = 1.1875

Factor a perfect square on the left side:
(x + -0.75)(x + -0.75) = 1.1875

Calculate the square root of the right side: 1.089724736

Break this problem into two subproblems by setting 
(x + -0.75) equal to 1.089724736 and -1.089724736.


Subproblem 1
x + -0.75 = 1.089724736

Simplifying
x + -0.75 = 1.089724736

Reorder the terms:
-0.75 + x = 1.089724736

Solving
-0.75 + x = 1.089724736

Solving for variable 'x'.

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

Add '0.75' to each side of the equation.
-0.75 + 0.75 + x = 1.089724736 + 0.75

Combine like terms: -0.75 + 0.75 = 0.00
0.00 + x = 1.089724736 + 0.75
x = 1.089724736 + 0.75

Combine like terms: 1.089724736 + 0.75 = 1.839724736
x = 1.839724736

Simplifying
x = 1.839724736


Subproblem 2
x + -0.75 = -1.089724736

Simplifying
x + -0.75 = -1.089724736

Reorder the terms:
-0.75 + x = -1.089724736

Solving
-0.75 + x = -1.089724736

Solving for variable 'x'.

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

Add '0.75' to each side of the equation.
-0.75 + 0.75 + x = -1.089724736 + 0.75

Combine like terms: -0.75 + 0.75 = 0.00
0.00 + x = -1.089724736 + 0.75
x = -1.089724736 + 0.75

Combine like terms: -1.089724736 + 0.75 = -0.339724736
x = -0.339724736

Simplifying
x = -0.339724736


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.839724736, -0.339724736}

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