Let u and v be the solutions to \(3x^2 + 5x + 7 = 0.\) Find \(\frac{u}{v} + \frac{v}{u}.\)
We could use the quad formula.....but....by Vieta....here's an easier solution method.....
u + v = -5/3 ⇒ (u + v)^2 = 25/9 ⇒ u^2 + 2uv + v^2 = 25/9 ⇒ u^2 + v^2 = 25/9 - 2uv
uv = 7/3 ⇒ 2uv = 2(7/3)
u/ v + v/ u =
(u^2 + v^2) / uv =
( 25/9 - 2uv) / ( 7/3) =
(25/9 - 2 ( 7/3)) / (7/3) =
(25/9 - 14/3) / (7/3) =
(25/9 - 42/9) / (7/3) =
(-17/9) ( 3/7) =
(-17/7) ( 3/9) =
(-17/7) (1/3) =
-17/21