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Let u and v be the solutions to 3x^2 + 5x + 11 = 0. FInd u/v + v/u.

 May 7, 2022
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We can rewrite \(\large{{u \over v} + {v \over u} }\) as \(\large{{u^2 \over xy} + {v^2 \over xy}}\). Because these have common denominators, we can simplify furthur: \(\large{{x^2+y^2} \over {xy}}\). Now recall the identity: \(a^2+b^2=(a+b)^2-2ab\). We can then apply this here, to get: \(\large{{(u+v)^2-2uv} \over uv}\).

 

Using Vieta's, we know that \(u+v= -{ b\over a} = -{5 \over 3}\) and \(uv = {c \over a} = {11 \over 3} \)

 

Now, we have to substitute these values and simplify. 


Can you take it from here?

 May 7, 2022

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