2x(x-5)-3(x-4)=0

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Solution for 2x(x-5)-3(x-4)=0 equation: 


Simplifying
2x(x + -5) + -3(x + -4) = 0

Reorder the terms:
2x(-5 + x) + -3(x + -4) = 0
(-5 * 2x + x * 2x) + -3(x + -4) = 0
(-10x + 2x2) + -3(x + -4) = 0

Reorder the terms:
-10x + 2x2 + -3(-4 + x) = 0
-10x + 2x2 + (-4 * -3 + x * -3) = 0
-10x + 2x2 + (12 + -3x) = 0

Reorder the terms:
12 + -10x + -3x + 2x2 = 0

Combine like terms: -10x + -3x = -13x
12 + -13x + 2x2 = 0

Solving
12 + -13x + 2x2 = 0

Solving for variable 'x'.

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

Divide each side by '2'.
6 + -6.5x + x2 = 0

Move the constant term to the right:

Add '-6' to each side of the equation.
6 + -6.5x + -6 + x2 = 0 + -6

Reorder the terms:
6 + -6 + -6.5x + x2 = 0 + -6

Combine like terms: 6 + -6 = 0
0 + -6.5x + x2 = 0 + -6
-6.5x + x2 = 0 + -6

Combine like terms: 0 + -6 = -6
-6.5x + x2 = -6

The x term is -6.5x.  Take half its coefficient (-3.25).
Square it (10.5625) and add it to both sides.

Add '10.5625' to each side of the equation.
-6.5x + 10.5625 + x2 = -6 + 10.5625

Reorder the terms:
10.5625 + -6.5x + x2 = -6 + 10.5625

Combine like terms: -6 + 10.5625 = 4.5625
10.5625 + -6.5x + x2 = 4.5625

Factor a perfect square on the left side:
(x + -3.25)(x + -3.25) = 4.5625

Calculate the square root of the right side: 2.136000936

Break this problem into two subproblems by setting 
(x + -3.25) equal to 2.136000936 and -2.136000936.


Subproblem 1
x + -3.25 = 2.136000936

Simplifying
x + -3.25 = 2.136000936

Reorder the terms:
-3.25 + x = 2.136000936

Solving
-3.25 + x = 2.136000936

Solving for variable 'x'.

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

Add '3.25' to each side of the equation.
-3.25 + 3.25 + x = 2.136000936 + 3.25

Combine like terms: -3.25 + 3.25 = 0.00
0.00 + x = 2.136000936 + 3.25
x = 2.136000936 + 3.25

Combine like terms: 2.136000936 + 3.25 = 5.386000936
x = 5.386000936

Simplifying
x = 5.386000936


Subproblem 2
x + -3.25 = -2.136000936

Simplifying
x + -3.25 = -2.136000936

Reorder the terms:
-3.25 + x = -2.136000936

Solving
-3.25 + x = -2.136000936

Solving for variable 'x'.

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

Add '3.25' to each side of the equation.
-3.25 + 3.25 + x = -2.136000936 + 3.25

Combine like terms: -3.25 + 3.25 = 0.00
0.00 + x = -2.136000936 + 3.25
x = -2.136000936 + 3.25

Combine like terms: -2.136000936 + 3.25 = 1.113999064
x = 1.113999064

Simplifying
x = 1.113999064


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {5.386000936, 1.113999064}

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