x(2x-3)=15

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Solution for x(2x-3)=15 equation: 


Simplifying
x(2x + -3) = 15

Reorder the terms:
x(-3 + 2x) = 15
(-3 * x + 2x * x) = 15
(-3x + 2x2) = 15

Solving
-3x + 2x2 = 15

Solving for variable 'x'.

Reorder the terms:
-15 + -3x + 2x2 = 15 + -15

Combine like terms: 15 + -15 = 0
-15 + -3x + 2x2 = 0

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

Divide each side by '2'.
-7.5 + -1.5x + x2 = 0

Move the constant term to the right:

Add '7.5' to each side of the equation.
-7.5 + -1.5x + 7.5 + x2 = 0 + 7.5

Reorder the terms:
-7.5 + 7.5 + -1.5x + x2 = 0 + 7.5

Combine like terms: -7.5 + 7.5 = 0.0
0.0 + -1.5x + x2 = 0 + 7.5
-1.5x + x2 = 0 + 7.5

Combine like terms: 0 + 7.5 = 7.5
-1.5x + x2 = 7.5

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 = 7.5 + 0.5625

Reorder the terms:
0.5625 + -1.5x + x2 = 7.5 + 0.5625

Combine like terms: 7.5 + 0.5625 = 8.0625
0.5625 + -1.5x + x2 = 8.0625

Factor a perfect square on the left side:
(x + -0.75)(x + -0.75) = 8.0625

Calculate the square root of the right side: 2.839454173

Break this problem into two subproblems by setting 
(x + -0.75) equal to 2.839454173 and -2.839454173.


Subproblem 1
x + -0.75 = 2.839454173

Simplifying
x + -0.75 = 2.839454173

Reorder the terms:
-0.75 + x = 2.839454173

Solving
-0.75 + x = 2.839454173

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 = 2.839454173 + 0.75

Combine like terms: -0.75 + 0.75 = 0.00
0.00 + x = 2.839454173 + 0.75
x = 2.839454173 + 0.75

Combine like terms: 2.839454173 + 0.75 = 3.589454173
x = 3.589454173

Simplifying
x = 3.589454173


Subproblem 2
x + -0.75 = -2.839454173

Simplifying
x + -0.75 = -2.839454173

Reorder the terms:
-0.75 + x = -2.839454173

Solving
-0.75 + x = -2.839454173

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 = -2.839454173 + 0.75

Combine like terms: -0.75 + 0.75 = 0.00
0.00 + x = -2.839454173 + 0.75
x = -2.839454173 + 0.75

Combine like terms: -2.839454173 + 0.75 = -2.089454173
x = -2.089454173

Simplifying
x = -2.089454173


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3.589454173, -2.089454173}

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