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# Algebra

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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