a(a-2b)+b(2b-a)=0

Solution for a(a-2b)+b(2b-a)=0 equation:

Simplifying
a(a + -2b) + b(2b + -1a) = 0
(a * a + -2b * a) + b(2b + -1a) = 0

Reorder the terms:
(-2ab + a2) + b(2b + -1a) = 0
(-2ab + a2) + b(2b + -1a) = 0

Reorder the terms:
-2ab + a2 + b(-1a + 2b) = 0
-2ab + a2 + (-1a * b + 2b * b) = 0
-2ab + a2 + (-1ab + 2b2) = 0

Reorder the terms:
-2ab + -1ab + a2 + 2b2 = 0

Combine like terms: -2ab + -1ab = -3ab
-3ab + a2 + 2b2 = 0

Solving
-3ab + a2 + 2b2 = 0

Solving for variable 'a'.

Factor a trinomial.
(a + -2b)(a + -1b) = 0

Subproblem 1
Set the factor '(a + -2b)' equal to zero and attempt to solve:

Simplifying
a + -2b = 0

Solving
a + -2b = 0

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

Add '2b' to each side of the equation.
a + -2b + 2b = 0 + 2b

Combine like terms: -2b + 2b = 0
a + 0 = 0 + 2b
a = 0 + 2b
Remove the zero:
a = 2b

Simplifying
a = 2b
Subproblem 2
Set the factor '(a + -1b)' equal to zero and attempt to solve:

Simplifying
a + -1b = 0

Solving
a + -1b = 0

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

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

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

Simplifying
a = b
Solution
a = {2b, b}