+1218 Let u and v be the solutions to 3x^2 + 5x + 7 = 2x^2 + 11x - 6. Find u/v^2 + v/u^2.
+1892 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
We already know all these terms! We can easily find them! We have
\(u^3 + v^3 = ( u + v) ( u^2 + v^2 - uv) = (6)(10 - 13) = -18 \\ [uv]^2 = [ 13]^2 = 169\)
So, The answer is \(u/v^2 + v/u^2 = [ u^3 + v^3 ] / [ uv]^2 = -18 / 169\)
Thanks! :)
