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Let  and  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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1) The roots of 3x^2 + 5x + 7 = 0 are -1/6*i*(sqrt(59) - 5i) and 1/6*i*(sqrt(59) = 5i).  Pugging these in, we get u/v + v/u = -19/21.

 

2) The roots of 7x^2 + x - 5 = 0 are 1/14 (-1 - sqrt(141)) and 1/14 (sqrt(141) - 1).  Plugging these in, we get (a - 4)(b - 4) = 113/7.

 

3) x - y = 4*sqrt(7).

 Mar 10, 2021

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