(ax-by)(ax+by)=0

Solution for (ax-by)(ax+by)=0 equation:

Simplifying
(ax + -1by)(ax + by) = 0

Multiply (ax + -1by) * (ax + by)
(ax(ax + by) + -1by * (ax + by)) = 0
((ax * ax + by * ax) + -1by * (ax + by)) = 0

Reorder the terms:
((abxy + a2x2) + -1by * (ax + by)) = 0
((abxy + a2x2) + -1by * (ax + by)) = 0
(abxy + a2x2 + (ax * -1by + by * -1by)) = 0
(abxy + a2x2 + (-1abxy + -1b2y2)) = 0

Reorder the terms:
(abxy + -1abxy + a2x2 + -1b2y2) = 0

Combine like terms: abxy + -1abxy = 0
(0 + a2x2 + -1b2y2) = 0
(a2x2 + -1b2y2) = 0

Solving
a2x2 + -1b2y2 = 0

Solving for variable 'a'.

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

Add 'b2y2' to each side of the equation.
a2x2 + -1b2y2 + b2y2 = 0 + b2y2

Combine like terms: -1b2y2 + b2y2 = 0
a2x2 + 0 = 0 + b2y2
a2x2 = 0 + b2y2
Remove the zero:
a2x2 = b2y2

Divide each side by 'x2'.
a2 = b2x-2y2

Simplifying
a2 = b2x-2y2

Take the square root of each side:
a = {-1bx-1y, bx-1y}