Question

If $a+b+c=0$, then write the value of $\frac{a^2}{bc}+\frac{b^2}{ca}+\frac{c^2}{ab}$

Answer 1

If $a+b+c=0$ then $a^3+b^3+c^3=3abc$

Given,

$a+b+c=0$, then $\frac{a^2}{bc}+\frac{b^2}{ca}+\frac{c^2}{ab}$

multiply and divide by, $a,b,c$

$=\frac{a}{a}\times \frac{a^2}{bc}+\frac{b}{b}\times \frac{b^2}{ca}+\frac{c}{c}\times \frac{c^2}{ab}$

$=\frac{a^3}{abc}+\frac{b^3}{abc}+\frac{c^3}{abc}$

$=\frac{a^3+b^3+c^3}{abc}$

but, $a^3+b^3+c^3=3abc$

$=\frac{3abc}{abc}$

$=3$