0.5x^2-8x+31.5=0

Solution for 0.5x^2-8x+31.5=0 equation:

Simplifying
0.5x2 + -8x + 31.5 = 0

Reorder the terms:
31.5 + -8x + 0.5x2 = 0

Solving
31.5 + -8x + 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'.
63 + -16x + x2 = 0

Move the constant term to the right:

Add '-63' to each side of the equation.
63 + -16x + -63 + x2 = 0 + -63

Reorder the terms:
63 + -63 + -16x + x2 = 0 + -63

Combine like terms: 63 + -63 = 0
0 + -16x + x2 = 0 + -63
-16x + x2 = 0 + -63

Combine like terms: 0 + -63 = -63
-16x + x2 = -63

The x term is -16x.  Take half its coefficient (-8).
Square it (64) and add it to both sides.

Add '64' to each side of the equation.
-16x + 64 + x2 = -63 + 64

Reorder the terms:
64 + -16x + x2 = -63 + 64

Combine like terms: -63 + 64 = 1
64 + -16x + x2 = 1

Factor a perfect square on the left side:
(x + -8)(x + -8) = 1

Calculate the square root of the right side: 1

Break this problem into two subproblems by setting 
(x + -8) equal to 1 and -1.

Subproblem 1
x + -8 = 1

Simplifying
x + -8 = 1

Reorder the terms:
-8 + x = 1

Solving
-8 + x = 1

Solving for variable 'x'.

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

Add '8' to each side of the equation.
-8 + 8 + x = 1 + 8

Combine like terms: -8 + 8 = 0
0 + x = 1 + 8
x = 1 + 8

Combine like terms: 1 + 8 = 9
x = 9

Simplifying
x = 9

Subproblem 2
x + -8 = -1

Simplifying
x + -8 = -1

Reorder the terms:
-8 + x = -1

Solving
-8 + x = -1

Solving for variable 'x'.

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

Add '8' to each side of the equation.
-8 + 8 + x = -1 + 8

Combine like terms: -8 + 8 = 0
0 + x = -1 + 8
x = -1 + 8

Combine like terms: -1 + 8 = 7
x = 7

Simplifying
x = 7

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {9, 7}