32Y^2-40Y+4=0

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Solution for 32Y^2-40Y+4=0 equation: 


Simplifying
32Y2 + -40Y + 4 = 0

Reorder the terms:
4 + -40Y + 32Y2 = 0

Solving
4 + -40Y + 32Y2 = 0

Solving for variable 'Y'.

Factor out the Greatest Common Factor (GCF), '4'.
4(1 + -10Y + 8Y2) = 0

Ignore the factor 4.

Subproblem 1
Set the factor '(1 + -10Y + 8Y2)' equal to zero and attempt to solve:

Simplifying
1 + -10Y + 8Y2 = 0

Solving
1 + -10Y + 8Y2 = 0

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

Divide each side by '8'.
0.125 + -1.25Y + Y2 = 0

Move the constant term to the right:

Add '-0.125' to each side of the equation.
0.125 + -1.25Y + -0.125 + Y2 = 0 + -0.125

Reorder the terms:
0.125 + -0.125 + -1.25Y + Y2 = 0 + -0.125

Combine like terms: 0.125 + -0.125 = 0.000
0.000 + -1.25Y + Y2 = 0 + -0.125
-1.25Y + Y2 = 0 + -0.125

Combine like terms: 0 + -0.125 = -0.125
-1.25Y + Y2 = -0.125

The Y term is -1.25Y.  Take half its coefficient (-0.625).
Square it (0.390625) and add it to both sides.

Add '0.390625' to each side of the equation.
-1.25Y + 0.390625 + Y2 = -0.125 + 0.390625

Reorder the terms:
0.390625 + -1.25Y + Y2 = -0.125 + 0.390625

Combine like terms: -0.125 + 0.390625 = 0.265625
0.390625 + -1.25Y + Y2 = 0.265625

Factor a perfect square on the left side:
(Y + -0.625)(Y + -0.625) = 0.265625

Calculate the square root of the right side: 0.515388203

Break this problem into two subproblems by setting 
(Y + -0.625) equal to 0.515388203 and -0.515388203.


Subproblem 1
Y + -0.625 = 0.515388203

Simplifying
Y + -0.625 = 0.515388203

Reorder the terms:
-0.625 + Y = 0.515388203

Solving
-0.625 + Y = 0.515388203

Solving for variable 'Y'.

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

Add '0.625' to each side of the equation.
-0.625 + 0.625 + Y = 0.515388203 + 0.625

Combine like terms: -0.625 + 0.625 = 0.000
0.000 + Y = 0.515388203 + 0.625
Y = 0.515388203 + 0.625

Combine like terms: 0.515388203 + 0.625 = 1.140388203
Y = 1.140388203

Simplifying
Y = 1.140388203


Subproblem 2
Y + -0.625 = -0.515388203

Simplifying
Y + -0.625 = -0.515388203

Reorder the terms:
-0.625 + Y = -0.515388203

Solving
-0.625 + Y = -0.515388203

Solving for variable 'Y'.

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

Add '0.625' to each side of the equation.
-0.625 + 0.625 + Y = -0.515388203 + 0.625

Combine like terms: -0.625 + 0.625 = 0.000
0.000 + Y = -0.515388203 + 0.625
Y = -0.515388203 + 0.625

Combine like terms: -0.515388203 + 0.625 = 0.109611797
Y = 0.109611797

Simplifying
Y = 0.109611797


Solution
The solution to the problem is based on the solutions
from the subproblems.
Y = {1.140388203, 0.109611797}
Solution
Y = {1.140388203, 0.109611797}

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