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Let u and v be the solutions to 3x^2+5x+7=2x^2-6x+1. Find u/v + v/u.

 Apr 28, 2022
 #1
avatar+2455 
+1

We can rewrite \(\large {{u \over v} + {v \over u}}\) as \(\large{{u^2+v^2} \over {uv}}\). We can also rewrite the numerator as: \(\large {{(u+v)^2-2uv} \over {uv}}\)

 

Using Vieta's, we know that \(u+v=-11\) and that \(uv=6\)

 

From here, substitute the values and simplify.

 

Feel free to ask if you need any help!!!

 Apr 29, 2022
 #2
avatar+13900 
+2

 Find u/v + v/u.

 

Hello Guest!

 

\(3x^2+5x+7=2x^2-6x+1\\ x^2+11x+6=0\\ x=-\frac{11}{2}\pm \sqrt{\frac{121}{4}-6}\\ \{u,v\}=\{ -5.5+\sqrt{\frac{97}{4}}, -5.5-\sqrt{\frac{97}{4}}\}\)

\(\dfrac{u}{v}=\frac{ -5.5+\sqrt{\frac{97}{4}} }{ -5.5-\sqrt{\frac{97}{4}}}=0.05521\)

\(\dfrac{v}{u}=18.111\)

\({\color{blue}\dfrac{u}{v}+\dfrac{v}{u}=}\frac{ -5.5+\sqrt{\frac{97}{4}} }{ -5.5-\sqrt{\frac{97}{4}}}+\frac{ -5.5-\sqrt{\frac{97}{4}} }{ -5.5+\sqrt{\frac{97}{4}}}=\color{blue}18.1\overline 6=\frac{109}{6}\)

laugh  !

 Apr 29, 2022
edited by asinus  Apr 29, 2022
 #3
avatar+124598 
+2

Rewrite as     x^2 + 11x  + 6  =  0

 

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

 

By Vieta

 

u +  v  =  -11             square both sides

u^2  + 2uv  + v^2    = 121

u^2 + v^2    =121   -  2uv       (1)

 

Also

uv =  6

So

2uv  = 12      (2)

 

Using (1)  and (2)

 

u/v  +  v/u  =     [ u^2  + v^2 ]  / [ uv]  =    [ 121 - 2uv ]  / [ uv]   =   [121 - 12]  /  6   =  109 /  6

 

 

cool cool cool

 Apr 29, 2022

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