0.5x(x+9)=5

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Solution for 0.5x(x+9)=5 equation: 


Simplifying
0.5x(x + 9) = 5

Reorder the terms:
0.5x(9 + x) = 5
(9 * 0.5x + x * 0.5x) = 5
(4.5x + 0.5x2) = 5

Solving
4.5x + 0.5x2 = 5

Solving for variable 'x'.

Reorder the terms:
-5 + 4.5x + 0.5x2 = 5 + -5

Combine like terms: 5 + -5 = 0
-5 + 4.5x + 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 + 9x + x2 = 0

Move the constant term to the right:

Add '10' to each side of the equation.
-10 + 9x + 10 + x2 = 0 + 10

Reorder the terms:
-10 + 10 + 9x + x2 = 0 + 10

Combine like terms: -10 + 10 = 0
0 + 9x + x2 = 0 + 10
9x + x2 = 0 + 10

Combine like terms: 0 + 10 = 10
9x + x2 = 10

The x term is 9x.  Take half its coefficient (4.5).
Square it (20.25) and add it to both sides.

Add '20.25' to each side of the equation.
9x + 20.25 + x2 = 10 + 20.25

Reorder the terms:
20.25 + 9x + x2 = 10 + 20.25

Combine like terms: 10 + 20.25 = 30.25
20.25 + 9x + x2 = 30.25

Factor a perfect square on the left side:
(x + 4.5)(x + 4.5) = 30.25

Calculate the square root of the right side: 5.5

Break this problem into two subproblems by setting 
(x + 4.5) equal to 5.5 and -5.5.


Subproblem 1
x + 4.5 = 5.5

Simplifying
x + 4.5 = 5.5

Reorder the terms:
4.5 + x = 5.5

Solving
4.5 + x = 5.5

Solving for variable 'x'.

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

Add '-4.5' to each side of the equation.
4.5 + -4.5 + x = 5.5 + -4.5

Combine like terms: 4.5 + -4.5 = 0.0
0.0 + x = 5.5 + -4.5
x = 5.5 + -4.5

Combine like terms: 5.5 + -4.5 = 1
x = 1

Simplifying
x = 1


Subproblem 2
x + 4.5 = -5.5

Simplifying
x + 4.5 = -5.5

Reorder the terms:
4.5 + x = -5.5

Solving
4.5 + x = -5.5

Solving for variable 'x'.

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

Add '-4.5' to each side of the equation.
4.5 + -4.5 + x = -5.5 + -4.5

Combine like terms: 4.5 + -4.5 = 0.0
0.0 + x = -5.5 + -4.5
x = -5.5 + -4.5

Combine like terms: -5.5 + -4.5 = -10
x = -10

Simplifying
x = -10


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1, -10}

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