Question Number 110268 by Khanacademy last updated on 28/Aug/20
Commented by som(math1967) last updated on 28/Aug/20
$$−\mathrm{2} \\ $$
Commented by Khanacademy last updated on 28/Aug/20
$$? \\ $$
Commented by som(math1967) last updated on 29/Aug/20
![((a^2 /(b+c))+a)+((b^2 /(c+a))+b)+((c^2 /(c+a))+c)−(a+b+c) ((a(a+b+c))/(b+c))+((b(a+b+c))/(c+a))+((c(a+b+c))/(a+b)) −(a+b+c) (a+b+c)((a/(b+c))+(b/(c+a))+(c/(a+b))) −(a+b+c) (a+b+c)(−1)−(a+b+c) [∵ (a/(b+c))+(b/(c+a))+(c/(a+b))=−1] −2(a+b+c) ∴((a^2 /(b+c))+(b^2 /(c+a))+(c^2 /(a+b))):(a+b+c) −2(a+b+c):(a+b+c) −2 [If (a+b+c)≠0]](https://www.tinkutara.com/question/Q110444.png)
$$\left(\frac{\mathrm{a}^{\mathrm{2}} }{\mathrm{b}+\mathrm{c}}+\mathrm{a}\right)+\left(\frac{\mathrm{b}^{\mathrm{2}} }{\mathrm{c}+\mathrm{a}}+\mathrm{b}\right)+\left(\frac{\mathrm{c}^{\mathrm{2}} }{\mathrm{c}+\mathrm{a}}+\mathrm{c}\right)−\left(\mathrm{a}+\mathrm{b}+\mathrm{c}\right) \\ $$$$\frac{\mathrm{a}\left(\mathrm{a}+\mathrm{b}+\mathrm{c}\right)}{\mathrm{b}+\mathrm{c}}+\frac{\mathrm{b}\left(\mathrm{a}+\mathrm{b}+\mathrm{c}\right)}{\mathrm{c}+\mathrm{a}}+\frac{\mathrm{c}\left(\mathrm{a}+\mathrm{b}+\mathrm{c}\right)}{\mathrm{a}+\mathrm{b}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:−\left(\mathrm{a}+\mathrm{b}+\mathrm{c}\right) \\ $$$$\left(\mathrm{a}+\mathrm{b}+\mathrm{c}\right)\left(\frac{\mathrm{a}}{\mathrm{b}+\mathrm{c}}+\frac{\mathrm{b}}{\mathrm{c}+\mathrm{a}}+\frac{\mathrm{c}}{\mathrm{a}+\mathrm{b}}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:−\left(\mathrm{a}+\mathrm{b}+\mathrm{c}\right) \\ $$$$\left(\mathrm{a}+\mathrm{b}+\mathrm{c}\right)\left(−\mathrm{1}\right)−\left(\mathrm{a}+\mathrm{b}+\mathrm{c}\right) \\ $$$$\left[\because\:\frac{\mathrm{a}}{\mathrm{b}+\mathrm{c}}+\frac{\mathrm{b}}{\mathrm{c}+\mathrm{a}}+\frac{\mathrm{c}}{\mathrm{a}+\mathrm{b}}=−\mathrm{1}\right] \\ $$$$−\mathrm{2}\left(\mathrm{a}+\mathrm{b}+\mathrm{c}\right) \\ $$$$\therefore\left(\frac{\mathrm{a}^{\mathrm{2}} }{\mathrm{b}+\mathrm{c}}+\frac{\mathrm{b}^{\mathrm{2}} }{\mathrm{c}+\mathrm{a}}+\frac{\mathrm{c}^{\mathrm{2}} }{\mathrm{a}+\mathrm{b}}\right):\left(\mathrm{a}+\mathrm{b}+\mathrm{c}\right) \\ $$$$−\mathrm{2}\left(\mathrm{a}+\mathrm{b}+\mathrm{c}\right):\left(\mathrm{a}+\mathrm{b}+\mathrm{c}\right) \\ $$$$−\mathrm{2}\:\:\:\:\left[\mathrm{If}\:\left(\mathrm{a}+\mathrm{b}+\mathrm{c}\right)\neq\mathrm{0}\right] \\ $$