32x^2+2xy-60y^2=0

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Solution for 32x^2+2xy-60y^2=0 equation: 


Simplifying
32x2 + 2xy + -60y2 = 0

Reorder the terms:
2xy + 32x2 + -60y2 = 0

Solving
2xy + 32x2 + -60y2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '2'.
2(xy + 16x2 + -30y2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(xy + 16x2 + -30y2)' equal to zero and attempt to solve:

Simplifying
xy + 16x2 + -30y2 = 0

Solving
xy + 16x2 + -30y2 = 0

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

Divide each side by '16'.
0.0625xy + x2 + -1.875y2 = 0

Move the constant term to the right:

Add '1.875y2' to each side of the equation.
0.0625xy + x2 + -1.875y2 + 1.875y2 = 0 + 1.875y2

Combine like terms: -1.875y2 + 1.875y2 = 0.000
0.0625xy + x2 + 0.000 = 0 + 1.875y2
0.0625xy + x2 = 0 + 1.875y2
Remove the zero:
0.0625xy + x2 = 1.875y2

The x term is xy.  Take half its coefficient (0.5y).
Square it (0.25y2) and add it to both sides.

Add '0.25y2' to each side of the equation.
0.0625xy + x2 + 0.25y2 = 1.875y2 + 0.25y2

Combine like terms: 1.875y2 + 0.25y2 = 2.125y2
0.0625xy + x2 + 0.25y2 = 2.125y2

Factor a perfect square on the left side:
(x + 0.5y)(x + 0.5y) = 2.125y2

Calculate the square root of the right side: 1.457737974y

Break this problem into two subproblems by setting 
(x + 0.5y) equal to 1.457737974y and -1.457737974y.


Subproblem 1
x + 0.5y = 1.457737974y

Simplifying
x + 0.5y = 1.457737974y

Solving
x + 0.5y = 1.457737974y

Solving for variable 'x'.

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

Add '-0.5y' to each side of the equation.
x + 0.5y + -0.5y = 1.457737974y + -0.5y

Combine like terms: 0.5y + -0.5y = 0.0
x + 0.0 = 1.457737974y + -0.5y
x = 1.457737974y + -0.5y

Combine like terms: 1.457737974y + -0.5y = 0.957737974y
x = 0.957737974y

Simplifying
x = 0.957737974y


Subproblem 2
x + 0.5y = -1.457737974y

Simplifying
x + 0.5y = -1.457737974y

Solving
x + 0.5y = -1.457737974y

Solving for variable 'x'.

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

Add '-0.5y' to each side of the equation.
x + 0.5y + -0.5y = -1.457737974y + -0.5y

Combine like terms: 0.5y + -0.5y = 0.0
x + 0.0 = -1.457737974y + -0.5y
x = -1.457737974y + -0.5y

Combine like terms: -1.457737974y + -0.5y = -1.957737974y
x = -1.957737974y

Simplifying
x = -1.957737974y


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.957737974y, -1.957737974y}
Solution
x = {0.957737974y, -1.957737974y}

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