18.92=-0.5x^2+1.5x+20

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Solution for 18.92=-0.5x^2+1.5x+20 equation: 


Simplifying
18.92 = -0.5x2 + 1.5x + 20

Reorder the terms:
18.92 = 20 + 1.5x + -0.5x2

Solving
18.92 = 20 + 1.5x + -0.5x2

Solving for variable 'x'.

Combine like terms: 18.92 + -20 = -1.08
-1.08 + -1.5x + 0.5x2 = 20 + 1.5x + -0.5x2 + -20 + -1.5x + 0.5x2

Reorder the terms:
-1.08 + -1.5x + 0.5x2 = 20 + -20 + 1.5x + -1.5x + -0.5x2 + 0.5x2

Combine like terms: 20 + -20 = 0
-1.08 + -1.5x + 0.5x2 = 0 + 1.5x + -1.5x + -0.5x2 + 0.5x2
-1.08 + -1.5x + 0.5x2 = 1.5x + -1.5x + -0.5x2 + 0.5x2

Combine like terms: 1.5x + -1.5x = 0.0
-1.08 + -1.5x + 0.5x2 = 0.0 + -0.5x2 + 0.5x2
-1.08 + -1.5x + 0.5x2 = -0.5x2 + 0.5x2

Combine like terms: -0.5x2 + 0.5x2 = 0.0
-1.08 + -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'.
-2.16 + -3x + x2 = 0

Move the constant term to the right:

Add '2.16' to each side of the equation.
-2.16 + -3x + 2.16 + x2 = 0 + 2.16

Reorder the terms:
-2.16 + 2.16 + -3x + x2 = 0 + 2.16

Combine like terms: -2.16 + 2.16 = 0.00
0.00 + -3x + x2 = 0 + 2.16
-3x + x2 = 0 + 2.16

Combine like terms: 0 + 2.16 = 2.16
-3x + x2 = 2.16

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 = 2.16 + 2.25

Reorder the terms:
2.25 + -3x + x2 = 2.16 + 2.25

Combine like terms: 2.16 + 2.25 = 4.41
2.25 + -3x + x2 = 4.41

Factor a perfect square on the left side:
(x + -1.5)(x + -1.5) = 4.41

Calculate the square root of the right side: 2.1

Break this problem into two subproblems by setting 
(x + -1.5) equal to 2.1 and -2.1.


Subproblem 1
x + -1.5 = 2.1

Simplifying
x + -1.5 = 2.1

Reorder the terms:
-1.5 + x = 2.1

Solving
-1.5 + x = 2.1

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 = 2.1 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + x = 2.1 + 1.5
x = 2.1 + 1.5

Combine like terms: 2.1 + 1.5 = 3.6
x = 3.6

Simplifying
x = 3.6


Subproblem 2
x + -1.5 = -2.1

Simplifying
x + -1.5 = -2.1

Reorder the terms:
-1.5 + x = -2.1

Solving
-1.5 + x = -2.1

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 = -2.1 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + x = -2.1 + 1.5
x = -2.1 + 1.5

Combine like terms: -2.1 + 1.5 = -0.6
x = -0.6

Simplifying
x = -0.6


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3.6, -0.6}

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