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if-b-2-3-c-2-2-a-b-c-a-2-calculate-a-b-c-




Question Number 126165 by TheMexicanTacosG last updated on 17/Dec/20
if:   b^2  = −3 −c^2  +2(a−b−c) −a^2     calculate: a + b + c = ?
$${if}:\:\:\:{b}^{\mathrm{2}} \:=\:−\mathrm{3}\:−{c}^{\mathrm{2}} \:+\mathrm{2}\left({a}−{b}−{c}\right)\:−{a}^{\mathrm{2}} \\ $$$$ \\ $$$${calculate}:\:{a}\:+\:{b}\:+\:{c}\:=\:? \\ $$
Commented by PRITHWISH SEN 2 last updated on 18/Dec/20
(a^2  −2a+1)+(b^2 +2b+1)+(c^2 +2c+1)=0   (a−1)^2 +(b+1)^2 +(c+1)^2 = 0  ⇒ a= 1, b=−1, c=−1  a+b+c=1−1−1=−1
$$\left(\mathrm{a}^{\mathrm{2}} \:−\mathrm{2a}+\mathrm{1}\right)+\left(\mathrm{b}^{\mathrm{2}} +\mathrm{2b}+\mathrm{1}\right)+\left(\mathrm{c}^{\mathrm{2}} +\mathrm{2c}+\mathrm{1}\right)=\mathrm{0}\: \\ $$$$\left(\mathrm{a}−\mathrm{1}\right)^{\mathrm{2}} +\left(\mathrm{b}+\mathrm{1}\right)^{\mathrm{2}} +\left(\mathrm{c}+\mathrm{1}\right)^{\mathrm{2}} =\:\mathrm{0} \\ $$$$\Rightarrow\:\boldsymbol{\mathrm{a}}=\:\mathrm{1},\:\boldsymbol{\mathrm{b}}=−\mathrm{1},\:\boldsymbol{\mathrm{c}}=−\mathrm{1} \\ $$$$\boldsymbol{\mathrm{a}}+\boldsymbol{\mathrm{b}}+\boldsymbol{\mathrm{c}}=\mathrm{1}−\mathrm{1}−\mathrm{1}=−\mathrm{1} \\ $$

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