If a^2 + b^2 + c^2 - ab - bc - ca =0, then: - Vidyadip

Question

If $a^2 + b^2 + c^2 - ab - bc - ca =0,$ then:

✓

Answer

$a^2 + b^2 + c^2 - ab - bc - ca = 0$
Multiplying by 2 on both the sides, we have
$2(a^2 + b^2 + c^2 - ab - bc - ca) = 0$
$2a^2+ 2b^2 + 2c^2 - 2ab - 2bc - 2ca = 0$
$a^2 + a^2 + b^2 + b^2+ c^2+ c^2 - 2ab - 2bc - 2ca = 0$
$(a^2 + b^2 - 2ab) + (b^2 + c^2 - 2bc) + (a^2 + c^2 - 2ac) = 0$
$(a - b)^2 + (b - c)^2 + (a - c)^2 = 0$
$(a - b)^2 = 0, (b - c)^2 = 0, (a - c)^2 = 0$
$(a - b) = 0, (b - c) = 0, (a - c) = 0$
$a = b, b = c, a = c$
or we can say $a = b = c$

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