mathmath333
  • mathmath333
fun question
Mathematics
jamiebookeater
  • jamiebookeater
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mathmath333
  • mathmath333
\(\large \color{black}{\begin{align} &\text{if}\quad p,q,r \quad \text{are roots of } \hspace{.33em}\\~\\ &x^3-5x-4=0 \hspace{.33em}\\~\\ &\text{find }\quad \left(\dfrac{1}{p+q}+\dfrac{1}{r+q}+\dfrac{1}{p+r}\right) \end{align}}\)
imqwerty
  • imqwerty
5/4
mathmath333
  • mathmath333
that's correct!

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anonymous
  • anonymous
yeah 5/4
mathmath333
  • mathmath333
by the way how did u find that
imqwerty
  • imqwerty
sorry for the cuttings
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ganeshie8
  • 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 \[\begin{align}& \frac{1}{-r}+\frac{1}{-p}+\frac{1}{-q}\\~\\&=-\frac{pq+qr+rp}{pqr}\\~\\&=-\frac{-5}{4}\end{align}\]
imqwerty
  • imqwerty
damn i didn't thought that way
skullpatrol
  • skullpatrol
you have two minus signs in your answer @ganeshie8
ganeshie8
  • ganeshie8
Haha final answer is indeed +5/4
skullpatrol
  • skullpatrol
:D
mathmath333
  • mathmath333
that was the actual way
skullpatrol
  • skullpatrol
what do you mean by "actual"?
mathmath333
  • mathmath333
efficient , optimal
skullpatrol
  • skullpatrol
Is there such a thing as an "efficient" or "optimal" way to have fun with a question in math?
ganeshie8
  • 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
imqwerty
  • imqwerty
that's what makes u the professor :)
mathmath333
  • 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
ParthKohli
  • ParthKohli
@parthkohli. holy hell, haha, who's this
skullpatrol
  • skullpatrol
Thank you for making the intent of your objective clear :-)

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