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Question-110268




Question Number 110268 by Khanacademy last updated on 28/Aug/20
Commented by som(math1967) last updated on 28/Aug/20
−2
$$−\mathrm{2} \\ $$
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]
$$\left(\frac{\mathrm{a}^{\mathrm{2}} }{\mathrm{b}+\mathrm{c}}+\mathrm{a}\right)+\left(\frac{\mathrm{b}^{\mathrm{2}} }{\mathrm{c}+\mathrm{a}}+\mathrm{b}\right)+\left(\frac{\mathrm{c}^{\mathrm{2}} }{\mathrm{c}+\mathrm{a}}+\mathrm{c}\right)−\left(\mathrm{a}+\mathrm{b}+\mathrm{c}\right) \\ $$$$\frac{\mathrm{a}\left(\mathrm{a}+\mathrm{b}+\mathrm{c}\right)}{\mathrm{b}+\mathrm{c}}+\frac{\mathrm{b}\left(\mathrm{a}+\mathrm{b}+\mathrm{c}\right)}{\mathrm{c}+\mathrm{a}}+\frac{\mathrm{c}\left(\mathrm{a}+\mathrm{b}+\mathrm{c}\right)}{\mathrm{a}+\mathrm{b}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:−\left(\mathrm{a}+\mathrm{b}+\mathrm{c}\right) \\ $$$$\left(\mathrm{a}+\mathrm{b}+\mathrm{c}\right)\left(\frac{\mathrm{a}}{\mathrm{b}+\mathrm{c}}+\frac{\mathrm{b}}{\mathrm{c}+\mathrm{a}}+\frac{\mathrm{c}}{\mathrm{a}+\mathrm{b}}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:−\left(\mathrm{a}+\mathrm{b}+\mathrm{c}\right) \\ $$$$\left(\mathrm{a}+\mathrm{b}+\mathrm{c}\right)\left(−\mathrm{1}\right)−\left(\mathrm{a}+\mathrm{b}+\mathrm{c}\right) \\ $$$$\left[\because\:\frac{\mathrm{a}}{\mathrm{b}+\mathrm{c}}+\frac{\mathrm{b}}{\mathrm{c}+\mathrm{a}}+\frac{\mathrm{c}}{\mathrm{a}+\mathrm{b}}=−\mathrm{1}\right] \\ $$$$−\mathrm{2}\left(\mathrm{a}+\mathrm{b}+\mathrm{c}\right) \\ $$$$\therefore\left(\frac{\mathrm{a}^{\mathrm{2}} }{\mathrm{b}+\mathrm{c}}+\frac{\mathrm{b}^{\mathrm{2}} }{\mathrm{c}+\mathrm{a}}+\frac{\mathrm{c}^{\mathrm{2}} }{\mathrm{a}+\mathrm{b}}\right):\left(\mathrm{a}+\mathrm{b}+\mathrm{c}\right) \\ $$$$−\mathrm{2}\left(\mathrm{a}+\mathrm{b}+\mathrm{c}\right):\left(\mathrm{a}+\mathrm{b}+\mathrm{c}\right) \\ $$$$−\mathrm{2}\:\:\:\:\left[\mathrm{If}\:\left(\mathrm{a}+\mathrm{b}+\mathrm{c}\right)\neq\mathrm{0}\right] \\ $$

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