4.6=-.5x^2+1.25x+4.5

Solution for 4.6=-.5x^2+1.25x+4.5 equation:

Simplifying
4.6 = -0.5x2 + 1.25x + 4.5

Reorder the terms:
4.6 = 4.5 + 1.25x + -0.5x2

Solving
4.6 = 4.5 + 1.25x + -0.5x2

Solving for variable 'x'.

Combine like terms: 4.6 + -4.5 = 0.1
0.1 + -1.25x + 0.5x2 = 4.5 + 1.25x + -0.5x2 + -4.5 + -1.25x + 0.5x2

Reorder the terms:
0.1 + -1.25x + 0.5x2 = 4.5 + -4.5 + 1.25x + -1.25x + -0.5x2 + 0.5x2

Combine like terms: 4.5 + -4.5 = 0.0
0.1 + -1.25x + 0.5x2 = 0.0 + 1.25x + -1.25x + -0.5x2 + 0.5x2
0.1 + -1.25x + 0.5x2 = 1.25x + -1.25x + -0.5x2 + 0.5x2

Combine like terms: 1.25x + -1.25x = 0.00
0.1 + -1.25x + 0.5x2 = 0.00 + -0.5x2 + 0.5x2
0.1 + -1.25x + 0.5x2 = -0.5x2 + 0.5x2

Combine like terms: -0.5x2 + 0.5x2 = 0.0
0.1 + -1.25x + 0.5x2 = 0.0

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

Divide each side by '0.5'.
0.2 + -2.5x + x2 = 0

Move the constant term to the right:

Add '-0.2' to each side of the equation.
0.2 + -2.5x + -0.2 + x2 = 0 + -0.2

Reorder the terms:
0.2 + -0.2 + -2.5x + x2 = 0 + -0.2

Combine like terms: 0.2 + -0.2 = 0.0
0.0 + -2.5x + x2 = 0 + -0.2
-2.5x + x2 = 0 + -0.2

Combine like terms: 0 + -0.2 = -0.2
-2.5x + x2 = -0.2

The x term is -2.5x.  Take half its coefficient (-1.25).
Square it (1.5625) and add it to both sides.

Add '1.5625' to each side of the equation.
-2.5x + 1.5625 + x2 = -0.2 + 1.5625

Reorder the terms:
1.5625 + -2.5x + x2 = -0.2 + 1.5625

Combine like terms: -0.2 + 1.5625 = 1.3625
1.5625 + -2.5x + x2 = 1.3625

Factor a perfect square on the left side:
(x + -1.25)(x + -1.25) = 1.3625

Calculate the square root of the right side: 1.167261753

Break this problem into two subproblems by setting 
(x + -1.25) equal to 1.167261753 and -1.167261753.

Subproblem 1
x + -1.25 = 1.167261753

Simplifying
x + -1.25 = 1.167261753

Reorder the terms:
-1.25 + x = 1.167261753

Solving
-1.25 + x = 1.167261753

Solving for variable 'x'.

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

Add '1.25' to each side of the equation.
-1.25 + 1.25 + x = 1.167261753 + 1.25

Combine like terms: -1.25 + 1.25 = 0.00
0.00 + x = 1.167261753 + 1.25
x = 1.167261753 + 1.25

Combine like terms: 1.167261753 + 1.25 = 2.417261753
x = 2.417261753

Simplifying
x = 2.417261753

Subproblem 2
x + -1.25 = -1.167261753

Simplifying
x + -1.25 = -1.167261753

Reorder the terms:
-1.25 + x = -1.167261753

Solving
-1.25 + x = -1.167261753

Solving for variable 'x'.

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

Add '1.25' to each side of the equation.
-1.25 + 1.25 + x = -1.167261753 + 1.25

Combine like terms: -1.25 + 1.25 = 0.00
0.00 + x = -1.167261753 + 1.25
x = -1.167261753 + 1.25

Combine like terms: -1.167261753 + 1.25 = 0.082738247
x = 0.082738247

Simplifying
x = 0.082738247

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.417261753, 0.082738247}