+2668 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!!!
+15001 Find u/v + v/u.
Hello Guest!
\(3x^2+5x+7=2x^2-6x+1\\ x^2+11x+6=0\\ x=-\frac{11}{2}\pm \sqrt{\frac{121}{4}-6}\\ \{u,v\}=\{ -5.5+\sqrt{\frac{97}{4}}, -5.5-\sqrt{\frac{97}{4}}\}\)
\(\dfrac{u}{v}=\frac{ -5.5+\sqrt{\frac{97}{4}} }{ -5.5-\sqrt{\frac{97}{4}}}=0.05521\)
\(\dfrac{v}{u}=18.111\)
\({\color{blue}\dfrac{u}{v}+\dfrac{v}{u}=}\frac{ -5.5+\sqrt{\frac{97}{4}} }{ -5.5-\sqrt{\frac{97}{4}}}+\frac{ -5.5-\sqrt{\frac{97}{4}} }{ -5.5+\sqrt{\frac{97}{4}}}=\color{blue}18.1\overline 6=\frac{109}{6}\)
!
+130077 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
