+645 Let u and v be the solutions to 3x^2 + 5x + 7 = 2x^2 + 11x - 6. Find u/v^2 + v/u^2.
+1876 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! :)
