8x=0.5x^2+31.5

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


Simplifying
8x = 0.5x2 + 31.5

Reorder the terms:
8x = 31.5 + 0.5x2

Solving
8x = 31.5 + 0.5x2

Solving for variable 'x'.

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

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

Combine like terms: 31.5 + -31.5 = 0.0
-31.5 + 8x + -0.5x2 = 0.0 + 0.5x2 + -0.5x2
-31.5 + 8x + -0.5x2 = 0.5x2 + -0.5x2

Combine like terms: 0.5x2 + -0.5x2 = 0.0
-31.5 + 8x + -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'.
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}

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