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# Help!

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1. Let a and b be the solutions to $$5x^2 - 11x + 4 = 0.$$ Find $$\frac{1}{a} + \frac{1}{b}.$$

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

Jan 22, 2021

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Hi Guest!

Vieta's Formula

To solve these problems we're going to need the vieta's formula.

Here's a link to them if you need them:

https://artofproblemsolving.com/wiki/index.php/Vieta%27s_Formulas

Problem 1

1/a + 1/b = (a+b)/(a*b)

a + b = 4/5 (Veita's)

a * b = 11/5 (Veita's)

(4/5) * (11/5) = 44/25

Problem 2

u/v + v/u = (u^2+v^2)/(u*v)

u^2+v^2 = (u+v)^2 - 2uv

u + v = 7/3

u * v = -7/5

u^2 + v^2 = (7/3)^2 - -14/5

u^2 + v^2 = 49/9 + 14/5

u^2 + v^2 = 371/45

u/v + v/u = (371/45)/(-7/5)

u/v + v/u = -53/9