x^2+4.5x+5=0

Solution for x^2+4.5x+5=0 equation:

Simplifying
x2 + 4.5x + 5 = 0

Reorder the terms:
5 + 4.5x + x2 = 0

Solving
5 + 4.5x + x2 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '-5' to each side of the equation.
5 + 4.5x + -5 + x2 = 0 + -5

Reorder the terms:
5 + -5 + 4.5x + x2 = 0 + -5

Combine like terms: 5 + -5 = 0
0 + 4.5x + x2 = 0 + -5
4.5x + x2 = 0 + -5

Combine like terms: 0 + -5 = -5
4.5x + x2 = -5

The x term is 4.5x.  Take half its coefficient (2.25).
Square it (5.0625) and add it to both sides.

Add '5.0625' to each side of the equation.
4.5x + 5.0625 + x2 = -5 + 5.0625

Reorder the terms:
5.0625 + 4.5x + x2 = -5 + 5.0625

Combine like terms: -5 + 5.0625 = 0.0625
5.0625 + 4.5x + x2 = 0.0625

Factor a perfect square on the left side:
(x + 2.25)(x + 2.25) = 0.0625

Calculate the square root of the right side: 0.25

Break this problem into two subproblems by setting 
(x + 2.25) equal to 0.25 and -0.25.

Subproblem 1
x + 2.25 = 0.25

Simplifying
x + 2.25 = 0.25

Reorder the terms:
2.25 + x = 0.25

Solving
2.25 + x = 0.25

Solving for variable 'x'.

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

Add '-2.25' to each side of the equation.
2.25 + -2.25 + x = 0.25 + -2.25

Combine like terms: 2.25 + -2.25 = 0.00
0.00 + x = 0.25 + -2.25
x = 0.25 + -2.25

Combine like terms: 0.25 + -2.25 = -2
x = -2

Simplifying
x = -2

Subproblem 2
x + 2.25 = -0.25

Simplifying
x + 2.25 = -0.25

Reorder the terms:
2.25 + x = -0.25

Solving
2.25 + x = -0.25

Solving for variable 'x'.

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

Add '-2.25' to each side of the equation.
2.25 + -2.25 + x = -0.25 + -2.25

Combine like terms: 2.25 + -2.25 = 0.00
0.00 + x = -0.25 + -2.25
x = -0.25 + -2.25

Combine like terms: -0.25 + -2.25 = -2.5
x = -2.5

Simplifying
x = -2.5

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-2, -2.5}