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 \(\frac{u}{v} + \frac{v}{u}.\)
+2407 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
Our answer is 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
Our answer is -53/9
*Please note that for the second problem, simply solving for the roots by factoring may be a better method.
I hope this helped. :))))
=^._.^=
