Question
If $a + b + c = 12$ and $a^2+ b^2+ c^2= 50;$ find $ab + bc + ca.$
✓
Answer
We know that
$(a+b+c)^2=a^2+b^2+c^2+2(a b+b c+c a) ........(1)$
Given that, $a^2+b^2+c^2=50$ and $a+b+c=12$.
We need to find $a b+b c+c a :$
Substitute the values of $\left(a^2+b^2+c^2\right)$ and $(a+b+c)$
in the identity $(1),$ we have
$(12)^2=50+2(a b+b c+c a)$
$ \Rightarrow 144=50+2(a b+b c+c a)$
$ \Rightarrow 94=2(a b+b c+c a)$
$ \Rightarrow a b+b c+c a=\frac{94}{2}$
$ \Rightarrow a b+b c+c a$
$=47$
$(a+b+c)^2=a^2+b^2+c^2+2(a b+b c+c a) ........(1)$
Given that, $a^2+b^2+c^2=50$ and $a+b+c=12$.
We need to find $a b+b c+c a :$
Substitute the values of $\left(a^2+b^2+c^2\right)$ and $(a+b+c)$
in the identity $(1),$ we have
$(12)^2=50+2(a b+b c+c a)$
$ \Rightarrow 144=50+2(a b+b c+c a)$
$ \Rightarrow 94=2(a b+b c+c a)$
$ \Rightarrow a b+b c+c a=\frac{94}{2}$
$ \Rightarrow a b+b c+c a$
$=47$
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