4Y^2=18Y-2

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Solution for 4Y^2=18Y-2 equation: 


Simplifying
4Y2 = 18Y + -2

Reorder the terms:
4Y2 = -2 + 18Y

Solving
4Y2 = -2 + 18Y

Solving for variable 'Y'.

Reorder the terms:
2 + -18Y + 4Y2 = -2 + 18Y + 2 + -18Y

Reorder the terms:
2 + -18Y + 4Y2 = -2 + 2 + 18Y + -18Y

Combine like terms: -2 + 2 = 0
2 + -18Y + 4Y2 = 0 + 18Y + -18Y
2 + -18Y + 4Y2 = 18Y + -18Y

Combine like terms: 18Y + -18Y = 0
2 + -18Y + 4Y2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(1 + -9Y + 2Y2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(1 + -9Y + 2Y2)' equal to zero and attempt to solve:

Simplifying
1 + -9Y + 2Y2 = 0

Solving
1 + -9Y + 2Y2 = 0

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

Divide each side by '2'.
0.5 + -4.5Y + Y2 = 0

Move the constant term to the right:

Add '-0.5' to each side of the equation.
0.5 + -4.5Y + -0.5 + Y2 = 0 + -0.5

Reorder the terms:
0.5 + -0.5 + -4.5Y + Y2 = 0 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + -4.5Y + Y2 = 0 + -0.5
-4.5Y + Y2 = 0 + -0.5

Combine like terms: 0 + -0.5 = -0.5
-4.5Y + Y2 = -0.5

The Y term is -4.5Y.  Take half its coefficient (-2.25).
Square it (5.0625) and add it to both sides.

Add '5.0625' to each side of the equation.
-4.5Y + 5.0625 + Y2 = -0.5 + 5.0625

Reorder the terms:
5.0625 + -4.5Y + Y2 = -0.5 + 5.0625

Combine like terms: -0.5 + 5.0625 = 4.5625
5.0625 + -4.5Y + Y2 = 4.5625

Factor a perfect square on the left side:
(Y + -2.25)(Y + -2.25) = 4.5625

Calculate the square root of the right side: 2.136000936

Break this problem into two subproblems by setting 
(Y + -2.25) equal to 2.136000936 and -2.136000936.


Subproblem 1
Y + -2.25 = 2.136000936

Simplifying
Y + -2.25 = 2.136000936

Reorder the terms:
-2.25 + Y = 2.136000936

Solving
-2.25 + Y = 2.136000936

Solving for variable 'Y'.

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

Add '2.25' to each side of the equation.
-2.25 + 2.25 + Y = 2.136000936 + 2.25

Combine like terms: -2.25 + 2.25 = 0.00
0.00 + Y = 2.136000936 + 2.25
Y = 2.136000936 + 2.25

Combine like terms: 2.136000936 + 2.25 = 4.386000936
Y = 4.386000936

Simplifying
Y = 4.386000936


Subproblem 2
Y + -2.25 = -2.136000936

Simplifying
Y + -2.25 = -2.136000936

Reorder the terms:
-2.25 + Y = -2.136000936

Solving
-2.25 + Y = -2.136000936

Solving for variable 'Y'.

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

Add '2.25' to each side of the equation.
-2.25 + 2.25 + Y = -2.136000936 + 2.25

Combine like terms: -2.25 + 2.25 = 0.00
0.00 + Y = -2.136000936 + 2.25
Y = -2.136000936 + 2.25

Combine like terms: -2.136000936 + 2.25 = 0.113999064
Y = 0.113999064

Simplifying
Y = 0.113999064


Solution
The solution to the problem is based on the solutions
from the subproblems.
Y = {4.386000936, 0.113999064}
Solution
Y = {4.386000936, 0.113999064}

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