0.5x^2+2x-2.5=0

Solution for 0.5x^2+2x-2.5=0 equation:

Simplifying
0.5x2 + 2x + -2.5 = 0

Reorder the terms:
-2.5 + 2x + 0.5x2 = 0

Solving
-2.5 + 2x + 0.5x2 = 0

Solving for variable 'x'.

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

Divide each side by '0.5'.
-5 + 4x + x2 = 0

Move the constant term to the right:

Add '5' to each side of the equation.
-5 + 4x + 5 + x2 = 0 + 5

Reorder the terms:
-5 + 5 + 4x + x2 = 0 + 5

Combine like terms: -5 + 5 = 0
0 + 4x + x2 = 0 + 5
4x + x2 = 0 + 5

Combine like terms: 0 + 5 = 5
4x + x2 = 5

The x term is 4x.  Take half its coefficient (2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
4x + 4 + x2 = 5 + 4

Reorder the terms:
4 + 4x + x2 = 5 + 4

Combine like terms: 5 + 4 = 9
4 + 4x + x2 = 9

Factor a perfect square on the left side:
(x + 2)(x + 2) = 9

Calculate the square root of the right side: 3

Break this problem into two subproblems by setting 
(x + 2) equal to 3 and -3.

Subproblem 1
x + 2 = 3

Simplifying
x + 2 = 3

Reorder the terms:
2 + x = 3

Solving
2 + x = 3

Solving for variable 'x'.

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

Add '-2' to each side of the equation.
2 + -2 + x = 3 + -2

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

Combine like terms: 3 + -2 = 1
x = 1

Simplifying
x = 1

Subproblem 2
x + 2 = -3

Simplifying
x + 2 = -3

Reorder the terms:
2 + x = -3

Solving
2 + x = -3

Solving for variable 'x'.

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

Add '-2' to each side of the equation.
2 + -2 + x = -3 + -2

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

Combine like terms: -3 + -2 = -5
x = -5

Simplifying
x = -5

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1, -5}