24x^2y+34xy+y=0

Solution for 24x^2y+34xy+y=0 equation:

Simplifying
24x2y + 34xy + y = 0

Reorder the terms:
34xy + 24x2y + y = 0

Solving
34xy + 24x2y + y = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), 'y'.
y(34x + 24x2 + 1) = 0

Subproblem 1
Set the factor 'y' equal to zero and attempt to solve:

Simplifying
y = 0

Solving
y = 0

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

Add '-1y' to each side of the equation.
y + -1y = 0 + -1y
Remove the zero:
0 = -1y

Simplifying
0 = -1y

The solution to this equation could not be determined.

This subproblem is being ignored because a solution could not be determined.
Subproblem 2
Set the factor '(34x + 24x2 + 1)' equal to zero and attempt to solve:

Simplifying
34x + 24x2 + 1 = 0

Reorder the terms:
1 + 34x + 24x2 = 0

Solving
1 + 34x + 24x2 = 0

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

Divide each side by '24'.
0.04166666667 + 1.416666667x + x2 = 0

Move the constant term to the right:

Add '-0.04166666667' to each side of the equation.
0.04166666667 + 1.416666667x + -0.04166666667 + x2 = 0 + -0.04166666667

Reorder the terms:
0.04166666667 + -0.04166666667 + 1.416666667x + x2 = 0 + -0.04166666667

Combine like terms: 0.04166666667 + -0.04166666667 = 0.00000000000
0.00000000000 + 1.416666667x + x2 = 0 + -0.04166666667
1.416666667x + x2 = 0 + -0.04166666667

Combine like terms: 0 + -0.04166666667 = -0.04166666667
1.416666667x + x2 = -0.04166666667

The x term is 1.416666667x.  Take half its coefficient (0.7083333335).
Square it (0.5017361113) and add it to both sides.

Add '0.5017361113' to each side of the equation.
1.416666667x + 0.5017361113 + x2 = -0.04166666667 + 0.5017361113

Reorder the terms:
0.5017361113 + 1.416666667x + x2 = -0.04166666667 + 0.5017361113

Combine like terms: -0.04166666667 + 0.5017361113 = 0.46006944463
0.5017361113 + 1.416666667x + x2 = 0.46006944463

Factor a perfect square on the left side:
(x + 0.7083333335)(x + 0.7083333335) = 0.46006944463

Calculate the square root of the right side: 0.678284192

Break this problem into two subproblems by setting 
(x + 0.7083333335) equal to 0.678284192 and -0.678284192.

Subproblem 1
x + 0.7083333335 = 0.678284192

Simplifying
x + 0.7083333335 = 0.678284192

Reorder the terms:
0.7083333335 + x = 0.678284192

Solving
0.7083333335 + x = 0.678284192

Solving for variable 'x'.

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

Add '-0.7083333335' to each side of the equation.
0.7083333335 + -0.7083333335 + x = 0.678284192 + -0.7083333335

Combine like terms: 0.7083333335 + -0.7083333335 = 0.0000000000
0.0000000000 + x = 0.678284192 + -0.7083333335
x = 0.678284192 + -0.7083333335

Combine like terms: 0.678284192 + -0.7083333335 = -0.0300491415
x = -0.0300491415

Simplifying
x = -0.0300491415

Subproblem 2
x + 0.7083333335 = -0.678284192

Simplifying
x + 0.7083333335 = -0.678284192

Reorder the terms:
0.7083333335 + x = -0.678284192

Solving
0.7083333335 + x = -0.678284192

Solving for variable 'x'.

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

Add '-0.7083333335' to each side of the equation.
0.7083333335 + -0.7083333335 + x = -0.678284192 + -0.7083333335

Combine like terms: 0.7083333335 + -0.7083333335 = 0.0000000000
0.0000000000 + x = -0.678284192 + -0.7083333335
x = -0.678284192 + -0.7083333335

Combine like terms: -0.678284192 + -0.7083333335 = -1.3866175255
x = -1.3866175255

Simplifying
x = -1.3866175255

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.0300491415, -1.3866175255}
Solution
x = {-0.0300491415, -1.3866175255}