0.25x^2+(1.25)(x)+1=0

Solution for 0.25x^2+(1.25)(x)+1=0 equation:

Simplifying
0.25x2 + (1.25)(x) + 1 = 0

Reorder the terms:
1 + 1.25x + 0.25x2 = 0

Solving
1 + 1.25x + 0.25x2 = 0

Solving for variable 'x'.

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

Divide each side by '0.25'.
4 + 5x + x2 = 0

Move the constant term to the right:

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

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

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

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

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 = -4 + 6.25

Reorder the terms:
6.25 + 5x + x2 = -4 + 6.25

Combine like terms: -4 + 6.25 = 2.25
6.25 + 5x + x2 = 2.25

Factor a perfect square on the left side:
(x + 2.5)(x + 2.5) = 2.25

Calculate the square root of the right side: 1.5

Break this problem into two subproblems by setting 
(x + 2.5) equal to 1.5 and -1.5.

Subproblem 1
x + 2.5 = 1.5

Simplifying
x + 2.5 = 1.5

Reorder the terms:
2.5 + x = 1.5

Solving
2.5 + x = 1.5

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 = 1.5 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + x = 1.5 + -2.5
x = 1.5 + -2.5

Combine like terms: 1.5 + -2.5 = -1
x = -1

Simplifying
x = -1

Subproblem 2
x + 2.5 = -1.5

Simplifying
x + 2.5 = -1.5

Reorder the terms:
2.5 + x = -1.5

Solving
2.5 + x = -1.5

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 = -1.5 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + x = -1.5 + -2.5
x = -1.5 + -2.5

Combine like terms: -1.5 + -2.5 = -4
x = -4

Simplifying
x = -4

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-1, -4}