How to solve this function

0 = 6x^2 + 512 / x

Where's my mistake?

-512 / x = 6x^2
-512 = 6x^3
-512 / 6 = x^3
(-512 / 6)^(1/3) = x
4.40 = x

Note, the answer must be positive because we're dealing with square feet.

Here's the original problem:

A rectangular storage container with an open top is to have a volume of 32 cubic feet. The length of the base is equal to the width of the base. Material for the base costs $6 per square foot and material for the sides cost $4 per square foot. e. Using the function C(x) (from part d), analytically find the WIDTH of the container that would produce the cheapest such container. Round your answer to two decimal places.

Your work is good up to C(x)= 6x^2+512/x

From there you assumed the cost could be zero.

Instead you are trying to find the value of x that will minimize C.

If you are in calculus, find C', then set =0. X= 3.49

If you do not know derivatives, graph the function, to find the minimum.(x=3.49)
when the cost will be $219.79

Hoping this helps!

your equation is...cost = C = 6X^2 + 512X^-1
differentiate and get dC/ dX = etc.......................= 0
solve for X .................solve for height

you do the simple differential.............