If $l+m+n=9$ and $l^2+m^2+n^2=31$, then find the value of $lm + mn + nl$.

Given: $l+m+n=9$ and $l^2+m^2+n^2=31$.

To do: To find the value of $lm + mn + nl$.

Solution:
 

As given, $l+m+n=9$ ......... $( i)$

$l^2+m^2+n^2=31$ ........ $( ii)$

On squaring equation $( i)$ both sides

$( l+m+n)^2=9^2$

$\Rightarrow l^2+2l( m+n)+( m+n)^2=81$

$\Rightarrow l^2+2lm+2ln+m^2+n^2+2mn=81$

$\Rightarrow l^2+m^2+n^2+2( lm+mn+nl)=81$

$\Rightarrow 31+2( lm+mn+nl)=81$

$\Rightarrow 2( lm+mn+nl)=81-31$

$\Rightarrow 2( lm+mn+nl)=50$

$\Rightarrow ( lm+mn+nl)=\frac{50}{2}$

$\Rightarrow ( lm+mn+nl)=25$

Thus, $( lm+mn+nl)=25$