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Let u and v be the solutions to 3x^2 + 5x + 7 = 2x^2 - 8x + 1.  Find u/v + v/u.

 Apr 26, 2021
 #1
avatar+25708 
+1

Let u and v be the solutions to 3x^2 + 5x + 7 = 2x^2 - 8x + 1.  Find u/v + v/u.

 

\(\begin{array}{|rcll|} \hline \mathbf{\dfrac{u}{v}+\dfrac{v}{u}} \\ &=& \dfrac{u^2+v^2}{uv} \\ &=& \dfrac{u^2+v^2}{uv} +2-2 \\ &=& \dfrac{u^2+v^2}{uv} +\dfrac{2uv}{uv}-2 \\ &=& \dfrac{u^2+2uv+v^2}{uv}-2 \\ &=& \dfrac{(u+v)^2}{uv}-2 \\ &=& \mathbf{\dfrac{(u+v)^2-2uv}{uv}} \\ \hline \mathbf{x^2+13x+6}&=&\mathbf{0} \\ \text{By Vieta: } \qquad u = -13, \quad v = 6 \\ \hline \end{array}\)

 

\(\begin{array}{|rcll|} \hline \mathbf{\dfrac{u}{v}+\dfrac{v}{u}} &=& \mathbf{\dfrac{(u+v)^2-2uv}{uv}} \\\\ &=& \dfrac{(-13)^2-2*6}{6} \\\\ &=& \dfrac{169-12}{6} \\\\ &=& \mathbf{\dfrac{157}{6}} \\ \hline \end{array} \)

 

laugh

 Apr 26, 2021
 #2
avatar+25708 
+2

Let u and v be the solutions to 3x^2 + 5x + 7 = 2x^2 - 8x + 1.  Find u/v + v/u.

 

\(\begin{array}{|rcll|} \hline \mathbf{\dfrac{u}{v}+\dfrac{v}{u}} \\ &=& \dfrac{u^2+v^2}{uv} \\\\ &=& \dfrac{u^2+v^2}{uv} +2-2 \\\\ &=& \dfrac{u^2+v^2}{uv} +\dfrac{2uv}{uv}-2 \\\\ &=& \dfrac{u^2+2uv+v^2}{uv}-2 \\\\ &=& \dfrac{(u+v)^2}{uv}-2 \\\\ \mathbf{\dfrac{u}{v}+\dfrac{v}{u}}&=& \mathbf{\dfrac{(u+v)^2-2uv}{uv}} \\ \hline \mathbf{x^2+13x+6}&=&\mathbf{0} \\ \text{By Vieta: } \qquad u+v = -13, \quad uv = 6 \\ \hline \end{array}\\ \begin{array}{|rcll|} \hline \mathbf{\dfrac{u}{v}+\dfrac{v}{u}} &=& \mathbf{\dfrac{(u+v)^2-2uv}{uv}} \quad | \quad u+v = -13, \quad uv = 6 \\\\ &=& \dfrac{(-13)^2-2*6}{6} \\\\ &=& \dfrac{169-12}{6} \\\\ &=& \mathbf{\dfrac{157}{6}} \\ \hline \end{array}\)

 

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heureka  Apr 26, 2021
edited by heureka  Apr 26, 2021

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