Question

fun question

Answer 1

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

Answer 2

5/4

Answer 3

that's correct!

Answer 4

yeah 5/4

Answer 5

by the way how did u find that

Answer 6

sorry for the cuttings

Answer 7

\[x^3-5x-4=0\]

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
\[\begin{align}& \frac{1}{-r}+\frac{1}{-p}+\frac{1}{-q}\\~\\&=-\frac{pq+qr+rp}{pqr}\\~\\&=-\frac{-5}{4}\end{align}\]

Answer 8

damn i didn't thought that way

Answer 9

you have two minus signs in your answer @ganeshie8

Answer 10

Haha final answer is indeed +5/4

Answer 11

:D

Answer 12

that was the actual way

Answer 13

what do you mean by "actual"?

Answer 14

efficient , optimal

Answer 15

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

Answer 16

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

Answer 17

that's what makes u the professor :)

Answer 18

@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

Answer 19

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

Answer 20

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