(2v-bu)(4v-bu)=0

Solution for (2v-bu)(4v-bu)=0 equation:

Simplifying
(2v + -1bu)(4v + -1bu) = 0

Reorder the terms:
(-1bu + 2v)(4v + -1bu) = 0

Reorder the terms:
(-1bu + 2v)(-1bu + 4v) = 0

Multiply (-1bu + 2v) * (-1bu + 4v)
(-1bu * (-1bu + 4v) + 2v * (-1bu + 4v)) = 0
((-1bu * -1bu + 4v * -1bu) + 2v * (-1bu + 4v)) = 0

Reorder the terms:
((-4buv + 1b2u2) + 2v * (-1bu + 4v)) = 0
((-4buv + 1b2u2) + 2v * (-1bu + 4v)) = 0
(-4buv + 1b2u2 + (-1bu * 2v + 4v * 2v)) = 0
(-4buv + 1b2u2 + (-2buv + 8v2)) = 0

Reorder the terms:
(-4buv + -2buv + 1b2u2 + 8v2) = 0

Combine like terms: -4buv + -2buv = -6buv
(-6buv + 1b2u2 + 8v2) = 0

Solving
-6buv + 1b2u2 + 8v2 = 0

Solving for variable 'b'.

Factor a trinomial.
(bu + -4v)(bu + -2v) = 0

Subproblem 1
Set the factor '(bu + -4v)' equal to zero and attempt to solve:

Simplifying
bu + -4v = 0

Solving
bu + -4v = 0

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

Add '4v' to each side of the equation.
bu + -4v + 4v = 0 + 4v

Combine like terms: -4v + 4v = 0
bu + 0 = 0 + 4v
bu = 0 + 4v
Remove the zero:
bu = 4v

Divide each side by 'u'.
b = 4u-1v

Simplifying
b = 4u-1v
Subproblem 2
Set the factor '(bu + -2v)' equal to zero and attempt to solve:

Simplifying
bu + -2v = 0

Solving
bu + -2v = 0

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

Add '2v' to each side of the equation.
bu + -2v + 2v = 0 + 2v

Combine like terms: -2v + 2v = 0
bu + 0 = 0 + 2v
bu = 0 + 2v
Remove the zero:
bu = 2v

Divide each side by 'u'.
b = 2u-1v

Simplifying
b = 2u-1v
Solution
b = {4u-1v, 2u-1v}