Finding Marshallian demands for Leontief production function with different powers Attempting to solve a practice problem, not sure if I'm heading in the right direction since my solution seems pretty messy. Given the following utility function,

$u(x,y)=min\{x^{1/2},2y\}$, find the Marshallian demands.

My answer:

Since Leontief is perfect complements, must be the case that $x^{1/2}=2y$, substituting this into a budget constraint yields the following:

$p_x \times x + p_y \times y = w$, where w is total income. Taking $x^{1/2}=2y$ and squaring this yields $x=4y^2$. Subbing this into constraint would give:

$p_x \times 4y^2 + p_y \times y = w$, at this point I applied the quadratic formula and got a demand function for y as follows,

$$y = \frac{-p_y \pm \sqrt{p_y^2 + 16 p_x w}}{8p_x}$$

This seems messy to me, I figure I can rule out the minus side of the quadratic seems that would imply y is negative. Even if some confirmation that this is the right approach would be appreciated.

Thanks!
As Amit said, your answer is correct. As Toby said, you can rule out negative output. 

There is no much insight to add as an answer really. However, I was also fairly puzzled that such a simple problem gives a complex answer. As it turns out, you get a linear demand (on income) only if the exponents on both components is the same. 

Consider the more general case:

$$ U(x,y) = \text{min}\{ax^\alpha,by^\beta\} $$

Optimality requires (* notation for optimal values omitted)

$$ ax^\alpha = by^\beta\ $$

Replacing one of these in the budget constraint leads to:

$$ p_x x + p_y \left(\frac{a}{b}\right)^{\frac{1}{\beta}} x^{\frac{\alpha}{\beta}} = w $$

Here we see that only when $\alpha=\beta$, demand will be linear (on income). In this case:

$$ x = \frac{w}{p_x + p_y\left(\frac{a}{b}\right)^{\frac{1}{\alpha}}} $$

It seems an algebraic solution for the general (unrestricted) case cannot be derived, as the polynomial you get might have irrational powers.