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

 Jun 22, 2024
 #1
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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! :)

 Jun 22, 2024

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