0.5x(x+3)=5

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Solution for 0.5x(x+3)=5 equation: 


Simplifying
0.5x(x + 3) = 5

Reorder the terms:
0.5x(3 + x) = 5
(3 * 0.5x + x * 0.5x) = 5
(1.5x + 0.5x2) = 5

Solving
1.5x + 0.5x2 = 5

Solving for variable 'x'.

Reorder the terms:
-5 + 1.5x + 0.5x2 = 5 + -5

Combine like terms: 5 + -5 = 0
-5 + 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'.
-10 + 3x + x2 = 0

Move the constant term to the right:

Add '10' to each side of the equation.
-10 + 3x + 10 + x2 = 0 + 10

Reorder the terms:
-10 + 10 + 3x + x2 = 0 + 10

Combine like terms: -10 + 10 = 0
0 + 3x + x2 = 0 + 10
3x + x2 = 0 + 10

Combine like terms: 0 + 10 = 10
3x + x2 = 10

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 = 10 + 2.25

Reorder the terms:
2.25 + 3x + x2 = 10 + 2.25

Combine like terms: 10 + 2.25 = 12.25
2.25 + 3x + x2 = 12.25

Factor a perfect square on the left side:
(x + 1.5)(x + 1.5) = 12.25

Calculate the square root of the right side: 3.5

Break this problem into two subproblems by setting 
(x + 1.5) equal to 3.5 and -3.5.


Subproblem 1
x + 1.5 = 3.5

Simplifying
x + 1.5 = 3.5

Reorder the terms:
1.5 + x = 3.5

Solving
1.5 + x = 3.5

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 = 3.5 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + x = 3.5 + -1.5
x = 3.5 + -1.5

Combine like terms: 3.5 + -1.5 = 2
x = 2

Simplifying
x = 2


Subproblem 2
x + 1.5 = -3.5

Simplifying
x + 1.5 = -3.5

Reorder the terms:
1.5 + x = -3.5

Solving
1.5 + x = -3.5

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 = -3.5 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + x = -3.5 + -1.5
x = -3.5 + -1.5

Combine like terms: -3.5 + -1.5 = -5
x = -5

Simplifying
x = -5


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2, -5}

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