(a-3b)(b-3a)=0

Solution for (a-3b)(b-3a)=0 equation:

Simplifying
(a + -3b)(b + -3a) = 0

Reorder the terms:
(a + -3b)(-3a + b) = 0

Multiply (a + -3b) * (-3a + b)
(a(-3a + b) + -3b * (-3a + b)) = 0
((-3a * a + b * a) + -3b * (-3a + b)) = 0

Reorder the terms:
((ab + -3a2) + -3b * (-3a + b)) = 0
((ab + -3a2) + -3b * (-3a + b)) = 0
(ab + -3a2 + (-3a * -3b + b * -3b)) = 0
(ab + -3a2 + (9ab + -3b2)) = 0

Reorder the terms:
(ab + 9ab + -3a2 + -3b2) = 0

Combine like terms: ab + 9ab = 10ab
(10ab + -3a2 + -3b2) = 0

Solving
10ab + -3a2 + -3b2 = 0

Solving for variable 'a'.

Factor a trinomial.
(-3a + b)(a + -3b) = 0

Subproblem 1
Set the factor '(-3a + b)' equal to zero and attempt to solve:

Simplifying
-3a + b = 0

Solving
-3a + b = 0

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

Add '-1b' to each side of the equation.
-3a + b + -1b = 0 + -1b

Combine like terms: b + -1b = 0
-3a + 0 = 0 + -1b
-3a = 0 + -1b
Remove the zero:
-3a = -1b

Divide each side by '-3'.
a = 0.3333333333b

Simplifying
a = 0.3333333333b
Subproblem 2
Set the factor '(a + -3b)' equal to zero and attempt to solve:

Simplifying
a + -3b = 0

Solving
a + -3b = 0

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

Add '3b' to each side of the equation.
a + -3b + 3b = 0 + 3b

Combine like terms: -3b + 3b = 0
a + 0 = 0 + 3b
a = 0 + 3b
Remove the zero:
a = 3b

Simplifying
a = 3b
Solution
a = {0.3333333333b, 3b}