a,b,c∈Z^+ and a^2 +b^2 +c^2 +ab+bc+ca=2025 find out all triplets (a,b,c).


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Question Number 217958 by dscm last updated on 23/Mar/25

$a, b, c \in Z^{+} a n d$ $a^{2} + b^{2} + c^{2} + a b + b c + c a = 2025$ $f i n d o u t a l l t r i p l e t s (a, b, c) .$
Commented by Unhombre last updated on 24/Mar/25

$y o u c a n a l s o u s e t h i s ↓$ $a^{2} + b^{2} + c^{2} + 2 a b + 2 a c + 2 b c =$ ${(a + b + c)}^{2}$
Commented by Frix last updated on 24/Mar/25

${(a + b + c)}^{2} = a^{2} + b^{2} + c^{2} + 2 (a b + a c + b c)$
	Answered by mehdee7396 last updated on 24/Mar/25

	$a + b = x & a + c = y & b + c = z$ $\Rightarrow x^{2} + y^{2} + z^{2} = 4050$ $\Rightarrow x = 40 & y = 35 & z = 35$ $\Rightarrow a = b = 20 & c = 15$
	Answered by mahdipoor last updated on 23/Mar/25

	${(a + b)}^{2} + {(b + c)}^{2} + {(c + a)}^{2} =$ $2 (a^{2} + b^{2} + c^{2} + a b + b c + c a) = 2 \times 2025$ $\Rightarrow 4050 = x^{2} + y^{2} + z^{2}$ $\Rightarrow x, y, z = . . .$ $\Rightarrow a, b, c = . . .$
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