<s>[INST] $p_1,p_2$ roots of $x^2+3x+5=0$, then the value of the expression $M=\frac{p_1^2+5p_1+7}{p_1^2+7p_1+5}+\frac{p_2^2+5p_2+7}{p_2^2+7p_2+5}$ If $p_1,p_2$ are the roots of the equation $x^2+3x+5=0$, then the value of the expression $M=\frac{p_1^2+5p_1+7}{p_1^2+7p_1+5}+\frac{p_2^2+5p_2+7}{p_2^2+7p_2+5}$
I solved the question as follows:
Initially I tried to solve it by finding what the roots are but since $b^2-4ac=9-20<0$, a real solution doesn't exist. Henceforth I attempted to solve it using Vieta's formulas, by saying that $p_1+p_2=-\frac{a_{n-1}}{a_n}=-\frac{3}{1}=-3$ and similarly $p_1*p_2=5$ and then $M=\frac{(p_1^2+5p_1+7)(p_2^2+7p_2+5)+(p_2^2+5p_2+7)(p_1^2+7p_1+5)}{(p_1^2+7p_1+5)(p_2^2+7p_2+5)}$
and eventually after a lot more calculations:
$M=\frac{7}{10}$
I find this method extremely laborious and I believe that there must exist a faster and easier method of solving this question. Could you please explain an easier and faster method?
 [/INST] Since $p_1,p_2$ are roots of $x^2+3x+5=0$, then $p_1^2+3p_1+5=p_2^2+3p2+5=0$ and we can simplify the expression to
$$
\frac{2p_1+2}{4p_1}+\frac{2p_2+2}{4p_2}=\frac{p_1+1}{2p_1}+\frac{p_2+1}{2p_2}=\frac{p_1p_2+p_2+p_1p_2+p_1}{2p_1p_2}=1+\frac{p_1+p_2}{2p_1p_2}
$$
Now $p_1+p_2=-3$ and $p_1p_2=5$ so $M=\frac{7}{10}$
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