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# Algebra

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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
+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
+2

Find u/v + v/u.

Hello Guest!

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

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

$$\dfrac{v}{u}=18.111$$

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

Apr 29, 2022
edited by asinus  Apr 29, 2022
#3
+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   Apr 29, 2022