15x-10=5x(x-2)

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Solution for 15x-10=5x(x-2) equation: 


Simplifying
15x + -10 = 5x(x + -2)

Reorder the terms:
-10 + 15x = 5x(x + -2)

Reorder the terms:
-10 + 15x = 5x(-2 + x)
-10 + 15x = (-2 * 5x + x * 5x)
-10 + 15x = (-10x + 5x2)

Solving
-10 + 15x = -10x + 5x2

Solving for variable 'x'.

Combine like terms: 15x + 10x = 25x
-10 + 25x + -5x2 = -10x + 5x2 + 10x + -5x2

Reorder the terms:
-10 + 25x + -5x2 = -10x + 10x + 5x2 + -5x2

Combine like terms: -10x + 10x = 0
-10 + 25x + -5x2 = 0 + 5x2 + -5x2
-10 + 25x + -5x2 = 5x2 + -5x2

Combine like terms: 5x2 + -5x2 = 0
-10 + 25x + -5x2 = 0

Factor out the Greatest Common Factor (GCF), '5'.
5(-2 + 5x + -1x2) = 0

Ignore the factor 5.

Subproblem 1
Set the factor '(-2 + 5x + -1x2)' equal to zero and attempt to solve:

Simplifying
-2 + 5x + -1x2 = 0

Solving
-2 + 5x + -1x2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
2 + -5x + x2 = 0

Move the constant term to the right:

Add '-2' to each side of the equation.
2 + -5x + -2 + x2 = 0 + -2

Reorder the terms:
2 + -2 + -5x + x2 = 0 + -2

Combine like terms: 2 + -2 = 0
0 + -5x + x2 = 0 + -2
-5x + x2 = 0 + -2

Combine like terms: 0 + -2 = -2
-5x + x2 = -2

The x term is -5x.  Take half its coefficient (-2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
-5x + 6.25 + x2 = -2 + 6.25

Reorder the terms:
6.25 + -5x + x2 = -2 + 6.25

Combine like terms: -2 + 6.25 = 4.25
6.25 + -5x + x2 = 4.25

Factor a perfect square on the left side:
(x + -2.5)(x + -2.5) = 4.25

Calculate the square root of the right side: 2.061552813

Break this problem into two subproblems by setting 
(x + -2.5) equal to 2.061552813 and -2.061552813.


Subproblem 1
x + -2.5 = 2.061552813

Simplifying
x + -2.5 = 2.061552813

Reorder the terms:
-2.5 + x = 2.061552813

Solving
-2.5 + x = 2.061552813

Solving for variable 'x'.

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

Add '2.5' to each side of the equation.
-2.5 + 2.5 + x = 2.061552813 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + x = 2.061552813 + 2.5
x = 2.061552813 + 2.5

Combine like terms: 2.061552813 + 2.5 = 4.561552813
x = 4.561552813

Simplifying
x = 4.561552813


Subproblem 2
x + -2.5 = -2.061552813

Simplifying
x + -2.5 = -2.061552813

Reorder the terms:
-2.5 + x = -2.061552813

Solving
-2.5 + x = -2.061552813

Solving for variable 'x'.

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

Add '2.5' to each side of the equation.
-2.5 + 2.5 + x = -2.061552813 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + x = -2.061552813 + 2.5
x = -2.061552813 + 2.5

Combine like terms: -2.061552813 + 2.5 = 0.438447187
x = 0.438447187

Simplifying
x = 0.438447187


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {4.561552813, 0.438447187}
Solution
x = {4.561552813, 0.438447187}

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