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Let u and v be the solutions to 3x^2 + 5x + 7 = 2x^2 + 9x - 17.  Find
\frac{u^2}{v} + \frac{v^2}{u}.

 Jan 6, 2024
 #1
avatar+126607 
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Note     u^2/v + v^2/u =   [u^3 + v^3] / (uv)

 

Simplify as

 

x^2 - 4x +  24  =  0

 

uv = 24

2uv = 48

 

u + v =  4           square both sides

u^2  + 2uv  + v^2  = 16

u^2 +48 + v^2  =16

u^2 + v^2  = -32

 

 

u^3  + v^3  =

(u + v)(u^2  - uv + v^2)

(u+ v) ( [u^2 + v^2] - uv)

(4) ( [ -32 ] - 24 ) 

(4) ( -56)  

-224

 

So 

 

u^2/v + v^2/u  = 

 

[u^3 + v^3 ]  / [uv ] =

 

-224 /  24  =    

 

-28 / 3

 

 

cool cool cool

 Jan 6, 2024

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