Algebra

First, let's simplify the equation. We have

$x^2 - 6x + 13$

 

Since the constant is equal to uv, we have the equation

$uv = 13 \\ 2uv = 26$

 

The coefficient of x is equal to u+v, so we have

$u + v = 6$

 

Squaring both sides of the equation, we have the equation

$u^2 + 2uv + v^2 = 36 \\ u^2 + 26 + v^2 = 36 \\ u^2 + v^2 = 10$

 

Now, it's time for the fun part of this problem. We can simplify what we want to figure out to

$u/v^2 + v/u^2 = [ u^3 + v^3 ] / [ uv]^2$

We already know all these terms! We can easily find them! We have

$u^3 + v^3 = ( u + v) ( u^2 + v^2 - uv) = (6)(10 - 13) = -18 \\ [uv]^2 = [ 13]^2 = 169$

 

Thus, our answer is $u/v^2 + v/u^2 = [ u^3 + v^3 ] / [ uv]^2 = -18 / 169$

 

So -18/169 is our answer. 

 

Thanks! :)