If a $^{2}$ + b $^{2}$ + c $^{2}$ = 2(a + 2b - 2c ) - 9 . Then ,find a + b + c

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Solution

Given $a^{2} + b^{2} + c^{2} = 2 (a + 2 b - 2 c) - 9$
$⟹ a^{2} - 2 a + b^{2} - 4 b + c^{2} + 4 c + 9 = 0$
$⟹ (a^{2} - 2 a + 1) + (b^{2} - 2 (b) (2) + 2^{2}) + (c^{2} + 2 (c) (2) + 2^{2}) = 0$
$⟹ (a - 1)^{2} + (b - 2)^{2} + (c + 2)^{2} = 0$
$⟹ a = 1, b = 2, c = - 2$
$a + b + c = 1 + 2 - 2 = 1$

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