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Let u and v be the solutions to $3x^2 + 5x + 7 = 0.$. Find \[\frac{u}{v} + \frac{v}{u}.\]

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The roots of $7x^2 + x - 5 = 0$ are a and b. Compute $(a - 4)(b - 4).$

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The real numbers x and y are such that  

\begin{align*}
x + y &= 4, \\
x^2 + y^2 &= 22, \\
x^4 &= y^4 - 176 \sqrt{7}.
\end{align*}

Compute x-y

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The roots of \[x^2 + 5x + 3 = 0\]

  are p and q, and the roots of  \[x^2 + bx + c = 0\] are $p^2$ and $q^2.$. Find b+c.

 Mar 10, 2021
 #1
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Try using Vieta!

 Mar 10, 2021
 #2
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For the first problem, where u and v are the roots, u + v = -5 and uv = 7 by Vieta.  Then

u/v + v/u = (u^2 + v^2)/(uv) = 11/7.

 

For the second problem, a + b = -1 and ab = -5.  Then

(a - 4)(b - 4) = ab - 4(a + b) + 16 = -5 - 4(-1) + 16 = 15.

 Mar 10, 2021

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