7x^2y+84xy-21y=0

Solution for 7x^2y+84xy-21y=0 equation:

Simplifying
7x2y + 84xy + -21y = 0

Reorder the terms:
84xy + 7x2y + -21y = 0

Solving
84xy + 7x2y + -21y = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '7y'.
7y(12x + x2 + -3) = 0

Ignore the factor 7.
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 '(12x + x2 + -3)' equal to zero and attempt to solve:

Simplifying
12x + x2 + -3 = 0

Reorder the terms:
-3 + 12x + x2 = 0

Solving
-3 + 12x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '3' to each side of the equation.
-3 + 12x + 3 + x2 = 0 + 3

Reorder the terms:
-3 + 3 + 12x + x2 = 0 + 3

Combine like terms: -3 + 3 = 0
0 + 12x + x2 = 0 + 3
12x + x2 = 0 + 3

Combine like terms: 0 + 3 = 3
12x + x2 = 3

The x term is 12x.  Take half its coefficient (6).
Square it (36) and add it to both sides.

Add '36' to each side of the equation.
12x + 36 + x2 = 3 + 36

Reorder the terms:
36 + 12x + x2 = 3 + 36

Combine like terms: 3 + 36 = 39
36 + 12x + x2 = 39

Factor a perfect square on the left side:
(x + 6)(x + 6) = 39

Calculate the square root of the right side: 6.244997998

Break this problem into two subproblems by setting 
(x + 6) equal to 6.244997998 and -6.244997998.

Subproblem 1
x + 6 = 6.244997998

Simplifying
x + 6 = 6.244997998

Reorder the terms:
6 + x = 6.244997998

Solving
6 + x = 6.244997998

Solving for variable 'x'.

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

Add '-6' to each side of the equation.
6 + -6 + x = 6.244997998 + -6

Combine like terms: 6 + -6 = 0
0 + x = 6.244997998 + -6
x = 6.244997998 + -6

Combine like terms: 6.244997998 + -6 = 0.244997998
x = 0.244997998

Simplifying
x = 0.244997998

Subproblem 2
x + 6 = -6.244997998

Simplifying
x + 6 = -6.244997998

Reorder the terms:
6 + x = -6.244997998

Solving
6 + x = -6.244997998

Solving for variable 'x'.

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

Add '-6' to each side of the equation.
6 + -6 + x = -6.244997998 + -6

Combine like terms: 6 + -6 = 0
0 + x = -6.244997998 + -6
x = -6.244997998 + -6

Combine like terms: -6.244997998 + -6 = -12.244997998
x = -12.244997998

Simplifying
x = -12.244997998

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.244997998, -12.244997998}
Solution
x = {0.244997998, -12.244997998}