2x^2-4x-22.5=0

Solution for 2x^2-4x-22.5=0 equation:

Simplifying
2x2 + -4x + -22.5 = 0

Reorder the terms:
-22.5 + -4x + 2x2 = 0

Solving
-22.5 + -4x + 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'.
-11.25 + -2x + x2 = 0

Move the constant term to the right:

Add '11.25' to each side of the equation.
-11.25 + -2x + 11.25 + x2 = 0 + 11.25

Reorder the terms:
-11.25 + 11.25 + -2x + x2 = 0 + 11.25

Combine like terms: -11.25 + 11.25 = 0.00
0.00 + -2x + x2 = 0 + 11.25
-2x + x2 = 0 + 11.25

Combine like terms: 0 + 11.25 = 11.25
-2x + x2 = 11.25

The x term is -2x.  Take half its coefficient (-1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
-2x + 1 + x2 = 11.25 + 1

Reorder the terms:
1 + -2x + x2 = 11.25 + 1

Combine like terms: 11.25 + 1 = 12.25
1 + -2x + x2 = 12.25

Factor a perfect square on the left side:
(x + -1)(x + -1) = 12.25

Calculate the square root of the right side: 3.5

Break this problem into two subproblems by setting 
(x + -1) equal to 3.5 and -3.5.

Subproblem 1
x + -1 = 3.5

Simplifying
x + -1 = 3.5

Reorder the terms:
-1 + x = 3.5

Solving
-1 + x = 3.5

Solving for variable 'x'.

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

Add '1' to each side of the equation.
-1 + 1 + x = 3.5 + 1

Combine like terms: -1 + 1 = 0
0 + x = 3.5 + 1
x = 3.5 + 1

Combine like terms: 3.5 + 1 = 4.5
x = 4.5

Simplifying
x = 4.5

Subproblem 2
x + -1 = -3.5

Simplifying
x + -1 = -3.5

Reorder the terms:
-1 + x = -3.5

Solving
-1 + x = -3.5

Solving for variable 'x'.

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

Add '1' to each side of the equation.
-1 + 1 + x = -3.5 + 1

Combine like terms: -1 + 1 = 0
0 + x = -3.5 + 1
x = -3.5 + 1

Combine like terms: -3.5 + 1 = -2.5
x = -2.5

Simplifying
x = -2.5

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {4.5, -2.5}