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Question Number 38679 by Tinkutara last updated on 28/Jun/18

	Answered by tanmay.chaudhury50@gmail.com last updated on 28/Jun/18

	$(a - x) (b c - b x - x c + x^{2} - a^{2}) - c (c^{2} - c x - a b)$ $+ b (a c - b^{2} + b x) = 0$ $a b c - a b x - a c x + {a x}^{2} - a^{3} - x b c + {b x}^{2} + {c x}^{2} - x^{3} +$ $a^{2} x - c^{3} + c^{2} x + a b c + a b c - b^{3} + b^{2} x = 0$ $3 a b c - a^{3} - b^{3} - c^{3} + x (a^{2} + b^{2} + c^{2} - a b - a c - b c) +$ $x^{2} (a + b + c) - x^{3} = 0$ $s i n c e a + b + c = 0 s o 3 a b c - a^{3} - b^{3} - c^{3} = 0$ $x (a^{2} + b^{2} + c^{2} - a b - b c - a c) - x^{3} = 0$ $n o w {(a + b + c)}^{2} = a^{2} + b^{2} + c^{2} + 2 (a b + b c + a c)$ $0 = a^{2} + b^{2} + c^{2} + 2 (a b + b c + a c)$ $- a b - b c - a c = \frac{a^{2} + b^{2} + c^{2}}{2}$ $s o t h e e q n i s$ $x (a^{2} + b^{2} + c^{2} + \frac{a^{2} + b^{2} + c^{2}}{2}) - x^{3} = 0$ $h e n c e e i t h e r x = 0 o r$ $\frac{3}{2} (a^{2} + b^{2} + c^{2}) - x^{2} = 0$ $x = \pm \sqrt{\frac{3}{2} (a^{2} + b^{2} + c^{2})}$ $p l e a z e n o t e e l e m e n t f i r s t r o w s e c o n d c o l l u m n$ ${()}_{1 \times 2} i s p r i n t e d a b u t i t s h l u l d b e c$
	Commented by Tinkutara last updated on 28/Jun/18
	Shouldn't we use properties of determinants to get a factorized expression?
	Commented by tanmay.chaudhury50@gmail.com last updated on 28/Jun/18

	$y e s . . l e t m e s o l v e b y a n o t h e r m e t h o d . . . u s i n g p r o p e r t i e s$
	Commented by tanmay.chaudhury50@gmail.com last updated on 04/Jul/18

	$p l s v i s i t a r c h i v e . o r g a n d d o w n l o a d$ $m a t h e m a t i c a l f o r m u l a a n d t a b l e s . . . o t h e r s$ $b o o k s . . . m u r r a y s p i e g e l . . . .$
	Commented by tanmay.chaudhury50@gmail.com last updated on 04/Jul/18

	Answered by tanmay.chaudhury50@gmail.com last updated on 28/Jun/18

	Answered by tanmay.chaudhury50@gmail.com last updated on 28/Jun/18

	Commented by Tinkutara last updated on 28/Jun/18
	Thank you very much Sir! I got the answer. ��
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