(3a+4b)*(3b-a)=0

Solution for (3a+4b)*(3b-a)=0 equation:

Simplifying
(3a + 4b)(3b + -1a) = 0

Reorder the terms:
(3a + 4b)(-1a + 3b) = 0

Multiply (3a + 4b) * (-1a + 3b)
(3a * (-1a + 3b) + 4b * (-1a + 3b)) = 0
((-1a * 3a + 3b * 3a) + 4b * (-1a + 3b)) = 0

Reorder the terms:
((9ab + -3a2) + 4b * (-1a + 3b)) = 0
((9ab + -3a2) + 4b * (-1a + 3b)) = 0
(9ab + -3a2 + (-1a * 4b + 3b * 4b)) = 0
(9ab + -3a2 + (-4ab + 12b2)) = 0

Reorder the terms:
(9ab + -4ab + -3a2 + 12b2) = 0

Combine like terms: 9ab + -4ab = 5ab
(5ab + -3a2 + 12b2) = 0

Solving
5ab + -3a2 + 12b2 = 0

Solving for variable 'a'.

Factor a trinomial.
(-3a + -4b)(a + -3b) = 0

Subproblem 1
Set the factor '(-3a + -4b)' equal to zero and attempt to solve:

Simplifying
-3a + -4b = 0

Solving
-3a + -4b = 0

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

Add '4b' to each side of the equation.
-3a + -4b + 4b = 0 + 4b

Combine like terms: -4b + 4b = 0
-3a + 0 = 0 + 4b
-3a = 0 + 4b
Remove the zero:
-3a = 4b

Divide each side by '-3'.
a = -1.333333333b

Simplifying
a = -1.333333333b
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 = {-1.333333333b, 3b}