3y+3y*3y=333333

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Solution for 3y+3y*3y=333333 equation: 


Simplifying
3y + 3y * 3y = 333333

Reorder the terms for easier multiplication:
3y + 3 * 3y * y = 333333

Multiply 3 * 3
3y + 9y * y = 333333

Multiply y * y
3y + 9y2 = 333333

Solving
3y + 9y2 = 333333

Solving for variable 'y'.

Reorder the terms:
-333333 + 3y + 9y2 = 333333 + -333333

Combine like terms: 333333 + -333333 = 0
-333333 + 3y + 9y2 = 0

Factor out the Greatest Common Factor (GCF), '3'.
3(-111111 + y + 3y2) = 0

Ignore the factor 3.

Subproblem 1
Set the factor '(-111111 + y + 3y2)' equal to zero and attempt to solve:

Simplifying
-111111 + y + 3y2 = 0

Solving
-111111 + y + 3y2 = 0

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

Divide each side by '3'.
-37037 + 0.3333333333y + y2 = 0

Move the constant term to the right:

Add '37037' to each side of the equation.
-37037 + 0.3333333333y + 37037 + y2 = 0 + 37037

Reorder the terms:
-37037 + 37037 + 0.3333333333y + y2 = 0 + 37037

Combine like terms: -37037 + 37037 = 0
0 + 0.3333333333y + y2 = 0 + 37037
0.3333333333y + y2 = 0 + 37037

Combine like terms: 0 + 37037 = 37037
0.3333333333y + y2 = 37037

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

Add '0.25' to each side of the equation.
0.3333333333y + 0.25 + y2 = 37037 + 0.25

Reorder the terms:
0.25 + 0.3333333333y + y2 = 37037 + 0.25

Combine like terms: 37037 + 0.25 = 37037.25
0.25 + 0.3333333333y + y2 = 37037.25

Factor a perfect square on the left side:
(y + 0.5)(y + 0.5) = 37037.25

Calculate the square root of the right side: 192.450643023

Break this problem into two subproblems by setting 
(y + 0.5) equal to 192.450643023 and -192.450643023.


Subproblem 1
y + 0.5 = 192.450643023

Simplifying
y + 0.5 = 192.450643023

Reorder the terms:
0.5 + y = 192.450643023

Solving
0.5 + y = 192.450643023

Solving for variable 'y'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + y = 192.450643023 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + y = 192.450643023 + -0.5
y = 192.450643023 + -0.5

Combine like terms: 192.450643023 + -0.5 = 191.950643023
y = 191.950643023

Simplifying
y = 191.950643023


Subproblem 2
y + 0.5 = -192.450643023

Simplifying
y + 0.5 = -192.450643023

Reorder the terms:
0.5 + y = -192.450643023

Solving
0.5 + y = -192.450643023

Solving for variable 'y'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + y = -192.450643023 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + y = -192.450643023 + -0.5
y = -192.450643023 + -0.5

Combine like terms: -192.450643023 + -0.5 = -192.950643023
y = -192.950643023

Simplifying
y = -192.950643023


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {191.950643023, -192.950643023}
Solution
y = {191.950643023, -192.950643023}

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