8.51=-0.5x^2+1.5x+9

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Solution for 8.51=-0.5x^2+1.5x+9 equation: 


Simplifying
8.51 = -0.5x2 + 1.5x + 9

Reorder the terms:
8.51 = 9 + 1.5x + -0.5x2

Solving
8.51 = 9 + 1.5x + -0.5x2

Solving for variable 'x'.

Combine like terms: 8.51 + -9 = -0.49
-0.49 + -1.5x + 0.5x2 = 9 + 1.5x + -0.5x2 + -9 + -1.5x + 0.5x2

Reorder the terms:
-0.49 + -1.5x + 0.5x2 = 9 + -9 + 1.5x + -1.5x + -0.5x2 + 0.5x2

Combine like terms: 9 + -9 = 0
-0.49 + -1.5x + 0.5x2 = 0 + 1.5x + -1.5x + -0.5x2 + 0.5x2
-0.49 + -1.5x + 0.5x2 = 1.5x + -1.5x + -0.5x2 + 0.5x2

Combine like terms: 1.5x + -1.5x = 0.0
-0.49 + -1.5x + 0.5x2 = 0.0 + -0.5x2 + 0.5x2
-0.49 + -1.5x + 0.5x2 = -0.5x2 + 0.5x2

Combine like terms: -0.5x2 + 0.5x2 = 0.0
-0.49 + -1.5x + 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'.
-0.98 + -3x + x2 = 0

Move the constant term to the right:

Add '0.98' to each side of the equation.
-0.98 + -3x + 0.98 + x2 = 0 + 0.98

Reorder the terms:
-0.98 + 0.98 + -3x + x2 = 0 + 0.98

Combine like terms: -0.98 + 0.98 = 0.00
0.00 + -3x + x2 = 0 + 0.98
-3x + x2 = 0 + 0.98

Combine like terms: 0 + 0.98 = 0.98
-3x + x2 = 0.98

The x term is -3x.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3x + 2.25 + x2 = 0.98 + 2.25

Reorder the terms:
2.25 + -3x + x2 = 0.98 + 2.25

Combine like terms: 0.98 + 2.25 = 3.23
2.25 + -3x + x2 = 3.23

Factor a perfect square on the left side:
(x + -1.5)(x + -1.5) = 3.23

Calculate the square root of the right side: 1.797220076

Break this problem into two subproblems by setting 
(x + -1.5) equal to 1.797220076 and -1.797220076.


Subproblem 1
x + -1.5 = 1.797220076

Simplifying
x + -1.5 = 1.797220076

Reorder the terms:
-1.5 + x = 1.797220076

Solving
-1.5 + x = 1.797220076

Solving for variable 'x'.

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

Add '1.5' to each side of the equation.
-1.5 + 1.5 + x = 1.797220076 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + x = 1.797220076 + 1.5
x = 1.797220076 + 1.5

Combine like terms: 1.797220076 + 1.5 = 3.297220076
x = 3.297220076

Simplifying
x = 3.297220076


Subproblem 2
x + -1.5 = -1.797220076

Simplifying
x + -1.5 = -1.797220076

Reorder the terms:
-1.5 + x = -1.797220076

Solving
-1.5 + x = -1.797220076

Solving for variable 'x'.

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

Add '1.5' to each side of the equation.
-1.5 + 1.5 + x = -1.797220076 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + x = -1.797220076 + 1.5
x = -1.797220076 + 1.5

Combine like terms: -1.797220076 + 1.5 = -0.297220076
x = -0.297220076

Simplifying
x = -0.297220076


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3.297220076, -0.297220076}

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