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