(3x+1.5)*x=220

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Solution for (3x+1.5)*x=220 equation: 


Simplifying
(3x + 1.5) * x = 220

Reorder the terms:
(1.5 + 3x) * x = 220

Reorder the terms for easier multiplication:
x(1.5 + 3x) = 220
(1.5 * x + 3x * x) = 220
(1.5x + 3x2) = 220

Solving
1.5x + 3x2 = 220

Solving for variable 'x'.

Reorder the terms:
-220 + 1.5x + 3x2 = 220 + -220

Combine like terms: 220 + -220 = 0
-220 + 1.5x + 3x2 = 0

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

Divide each side by '3'.
-73.33333333 + 0.5x + x2 = 0

Move the constant term to the right:

Add '73.33333333' to each side of the equation.
-73.33333333 + 0.5x + 73.33333333 + x2 = 0 + 73.33333333

Reorder the terms:
-73.33333333 + 73.33333333 + 0.5x + x2 = 0 + 73.33333333

Combine like terms: -73.33333333 + 73.33333333 = 0.00000000
0.00000000 + 0.5x + x2 = 0 + 73.33333333
0.5x + x2 = 0 + 73.33333333

Combine like terms: 0 + 73.33333333 = 73.33333333
0.5x + x2 = 73.33333333

The x term is 0.5x.  Take half its coefficient (0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
0.5x + 0.0625 + x2 = 73.33333333 + 0.0625

Reorder the terms:
0.0625 + 0.5x + x2 = 73.33333333 + 0.0625

Combine like terms: 73.33333333 + 0.0625 = 73.39583333
0.0625 + 0.5x + x2 = 73.39583333

Factor a perfect square on the left side:
(x + 0.25)(x + 0.25) = 73.39583333

Calculate the square root of the right side: 8.567136822

Break this problem into two subproblems by setting 
(x + 0.25) equal to 8.567136822 and -8.567136822.


Subproblem 1
x + 0.25 = 8.567136822

Simplifying
x + 0.25 = 8.567136822

Reorder the terms:
0.25 + x = 8.567136822

Solving
0.25 + x = 8.567136822

Solving for variable 'x'.

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

Add '-0.25' to each side of the equation.
0.25 + -0.25 + x = 8.567136822 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + x = 8.567136822 + -0.25
x = 8.567136822 + -0.25

Combine like terms: 8.567136822 + -0.25 = 8.317136822
x = 8.317136822

Simplifying
x = 8.317136822


Subproblem 2
x + 0.25 = -8.567136822

Simplifying
x + 0.25 = -8.567136822

Reorder the terms:
0.25 + x = -8.567136822

Solving
0.25 + x = -8.567136822

Solving for variable 'x'.

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

Add '-0.25' to each side of the equation.
0.25 + -0.25 + x = -8.567136822 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + x = -8.567136822 + -0.25
x = -8.567136822 + -0.25

Combine like terms: -8.567136822 + -0.25 = -8.817136822
x = -8.817136822

Simplifying
x = -8.817136822


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {8.317136822, -8.817136822}

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