## mathmath333 one year ago fun question

1. mathmath333

\large \color{black}{\begin{align} &\text{if}\quad p,q,r \quad \text{are roots of } \hspace{.33em}\\~\\ &x^3-5x-4=0 \hspace{.33em}\\~\\ &\text{find }\quad \left(\dfrac{1}{p+q}+\dfrac{1}{r+q}+\dfrac{1}{p+r}\right) \end{align}}

2. imqwerty

5/4

3. mathmath333

that's correct!

4. anonymous

yeah 5/4

5. mathmath333

by the way how did u find that

6. imqwerty

sorry for the cuttings

7. ganeshie8

$x^3-5x-4=0$ Vieta's formulas gives sum of roots : $$p+q+r=0 \implies p+q=-r, ~q+r=-p,~~r+p=-q$$ $$pq+qr+rp=-5$$ $$pqr=4$$ plugging them in the given expression we get \begin{align}& \frac{1}{-r}+\frac{1}{-p}+\frac{1}{-q}\\~\\&=-\frac{pq+qr+rp}{pqr}\\~\\&=-\frac{-5}{4}\end{align}

8. imqwerty

damn i didn't thought that way

9. skullpatrol

10. ganeshie8

Haha final answer is indeed +5/4

11. skullpatrol

:D

12. mathmath333

that was the actual way

13. skullpatrol

what do you mean by "actual"?

14. mathmath333

efficient , optimal

15. skullpatrol

Is there such a thing as an "efficient" or "optimal" way to have fun with a question in math?

16. ganeshie8

I think many times there exists "the" best way to solve a problem... initially i tried combining the fractions and expanding everything but that went v messy it actually took some time to see that p+q+r=0 is useful

17. imqwerty

that's what makes u the professor :)

18. mathmath333

@skullpatrol using the fact p+q+r=0 makes it easy , while trying to solve it with rigour loos like to kill a rat with atom bomb that's why it was fun question

19. ParthKohli

@parthkohli. holy hell, haha, who's this

20. skullpatrol

Thank you for making the intent of your objective clear :-)