if a+b+c =3 and a^3 +b^3 +c^3 =27 find a^2 +b^2 +c^2


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Question Number 72027 by mathmax by abdo last updated on 23/Oct/19

$i f a + b + c = 3 a n d a^{3} + b^{3} + c^{3} = 27 f i n d a^{2} + b^{2} + c^{2}$
Commented by Rasheed.Sindhi last updated on 26/Oct/19

$I n s u f f i c i e n t d a t a f o r u n i q u e s o l u t i o n .$ $T h e r e a r e t h r e e u n k n o w n s, s o t h r e e$ $e q u a t i o n s a r e r e q u i r e d f o r u n i q u e$ $s o l u t i o n .$
	Answered by Rasheed.Sindhi last updated on 26/Oct/19

	$F o r u n i q u e s o l u t i o n a t h i r d e q u a t i o n$ $i s n e c e s s a r y, b e c a u s e t h e r e a r e t h r e e$ $u n k n o w n s .$ $S i n c e t h e t h i r d e q u a t i o n i s n o t g i v e n$ $w e a s s u m e v a l u e o f o n e u n k n o w n :$ $L e t c = 1$ $a + b = 2, a^{3} + b^{3} = 26$ ${(a + b)}^{3} = 8$ $a^{3} + b^{3} + 3 a b (a + b) = 8$ $26 + 3 a b (2) = 8$ $a b = \frac{- 18}{6} = - 3$ $a (2 - a) = - 3$ $a^{2} - 2 a - 3 = 0$ $(a - 3) (a + 1) = 0$ $a = 3 \lor a = - 1$ $b = 2 - 3 = - 1 \lor b = 2 + 1 = 3$ $(a, b, c) = (3, - 1, 1)$ $o r$ $(a, b, c) = (- 1, 3, 1)$ $I f c = 1, - 1 o r 3 :$ $a^{2} + b^{2} + c^{2} = {(- 1)}^{2} + {(3)}^{2} + {(1)}^{2} = 6$
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