(.5)(x+3)(x)=20

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Solution for (.5)(x+3)(x)=20 equation: 


Simplifying
(0.5)(x + 3)(x) = 20

Reorder the terms:
0.5(3 + x)(x) = 20

Reorder the terms for easier multiplication:
0.5x(3 + x) = 20
(3 * 0.5x + x * 0.5x) = 20
(1.5x + 0.5x2) = 20

Solving
1.5x + 0.5x2 = 20

Solving for variable 'x'.

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

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

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

Divide each side by '0.5'.
-40 + 3x + x2 = 0

Move the constant term to the right:

Add '40' to each side of the equation.
-40 + 3x + 40 + x2 = 0 + 40

Reorder the terms:
-40 + 40 + 3x + x2 = 0 + 40

Combine like terms: -40 + 40 = 0
0 + 3x + x2 = 0 + 40
3x + x2 = 0 + 40

Combine like terms: 0 + 40 = 40
3x + x2 = 40

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 = 40 + 2.25

Reorder the terms:
2.25 + 3x + x2 = 40 + 2.25

Combine like terms: 40 + 2.25 = 42.25
2.25 + 3x + x2 = 42.25

Factor a perfect square on the left side:
(x + 1.5)(x + 1.5) = 42.25

Calculate the square root of the right side: 6.5

Break this problem into two subproblems by setting 
(x + 1.5) equal to 6.5 and -6.5.


Subproblem 1
x + 1.5 = 6.5

Simplifying
x + 1.5 = 6.5

Reorder the terms:
1.5 + x = 6.5

Solving
1.5 + x = 6.5

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 = 6.5 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + x = 6.5 + -1.5
x = 6.5 + -1.5

Combine like terms: 6.5 + -1.5 = 5
x = 5

Simplifying
x = 5


Subproblem 2
x + 1.5 = -6.5

Simplifying
x + 1.5 = -6.5

Reorder the terms:
1.5 + x = -6.5

Solving
1.5 + x = -6.5

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 = -6.5 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + x = -6.5 + -1.5
x = -6.5 + -1.5

Combine like terms: -6.5 + -1.5 = -8
x = -8

Simplifying
x = -8


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {5, -8}

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