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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.

Guest Mar 10, 2021

Guest · Answer 1 · 2021-03-10T02:25:14

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).