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Question Number 44729 by behi83417@gmail.com last updated on 03/Oct/18

	Answered by MJS last updated on 04/Oct/18

	$trivial solutions a = b = c = 0 \lor a = b = c = \frac{1}{3}$ $c^{2} + b c + b^{2} - a = 0$ $c^{2} + a c + a^{2} - b = 0$ $(b - a) c + b^{2} - a^{2} + b - a = 0$ $c = - (a + b + 1)$ $a^{2} + b^{2} + a b + a + b + 1 = 0$ $b^{2} + (a + 1) b + a^{2} + a + 1 = 0$ $b = - \frac{a + 1}{2} \pm \frac{\sqrt{- 3 a^{2} - 2 a - 3}}{2}$ $c = - \frac{a + 1}{2} \mp \frac{\sqrt{- 3 a^{2} - 2 a - 3}}{2}$ $a = b^{2} + c^{2} + b c always true$ $a \in R \Rightarrow b, c \notin R$
	Answered by ajfour last updated on 04/Oct/18

	$3 x^{2} = x$ $\Rightarrow a = b = c = \frac{1}{3} (x a b o v e) .$
	Answered by ajfour last updated on 04/Oct/18

	$b = c$ $a = 3 b^{2}$ $b = a^{2} + b^{2} + a b$ $b = 9 b^{4} + b^{2} + 3 b^{3}$ $\Rightarrow 9 b^{3} + 3 b^{2} + b = 1$ $\Rightarrow b = c = \frac{1}{3} = a .$
	Commented by behi83417@gmail.com last updated on 04/Oct/18

	$t h a n k y o u v e r y m u c h d e a r m y$ $f r i e n d s .$ $. . . . a l s o w a i t i n g f o r m r W 3^{'} s p r o o f .$
	Answered by MrW3 last updated on 04/Oct/18

	$(i) - (i i) :$ $a - b = b^{2} - a^{2} + b c - a c$ $(b + a) (b - a) + (b - a) c + (b - a) = 0$ $\Rightarrow (a + b + c + 1) (b - a) = 0$ $s i m i l a r l y$ $\Rightarrow (a + b + c + 1) (c - b) = 0$ $\Rightarrow (a + b + c + 1) (a - c) = 0$ $c a s e 1 : a = b = c = k$ $\Rightarrow k = 3 k^{2}$ $\Rightarrow k = 0 o r k = \frac{1}{3}$ $c a s e 2 : a + b + c + 1 = 0$ $a = {(b + c)}^{2} - b c . . . (1)$ $b = {(c + a)}^{2} - c a . . . (2)$ $c = {(a + b)}^{2} - a b . . . (3)$ $a = {(1 + a)}^{2} - b c$ $b = {(a + c)}^{2} - c a$ $c = {(1 + c)}^{2} - a b$ $a = {(1 + a)}^{2} - c [{(a + c)}^{2} - c a]$ $a = {(1 + a)}^{2} - c {(a + c)}^{2} + c^{2} a$ $a^{2} = a {(1 + a)}^{2} - a c {(a + c)}^{2} + c^{2} a^{2}$ $c = {(1 + c)}^{2} - a [{(a + c)}^{2} - c a]$ $c = {(1 + c)}^{2} - a {(a + c)}^{2} + a^{2} c$ $c^{2} = c {(1 + c)}^{2} - a c {(a + c)}^{2} + a^{2} c^{2}$ $a^{2} - c^{2} = a {(1 + a)}^{2} - c {(1 + c)}^{2}$ $a^{2} - c^{2} = a + 2 a^{2} + a^{3} - c - 2 c^{2} - c^{3}$ $a + a^{2} + a^{3} = c + c^{2} + c^{3} = h s a y$ $s i m i l a r l y$ $a + a^{2} + a^{3} = b + b^{2} + b^{3} = h$ $s i n c e x + x^{2} + x^{3} = h h a s o n l y o n e s i n g l e$ $r e a l s o l u t i o n, l e t s s a y x = p,$ $w e g e t a = b = c = x = p$ $i . e . c a s e 2 g i v e s t h e s a m e s o l u t i o n a s$ $c a s e 1 .$
	Commented by behi83417@gmail.com last updated on 04/Oct/18

	$p e r f e c t! t h a n k y o u d e a r m a s t e r .$
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