f(x)+f(2x+y)+5xy = f(3x−y)+x^2 +1 for every x,y∈R . find f(10)


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Question Number 109262 by bemath last updated on 22/Aug/20

$f (x) + f (2 x + y) + 5 x y = f (3 x - y) + x^{2} + 1$ $f o r e v e r y x, y \in R . f i n d f (10)$
	Answered by mr W last updated on 22/Aug/20

	$y = 0 :$ $f (x) + f (2 x) = f (3 x) + x^{2} + 1 . . . (i)$ $y = x :$ $f (x) + f (3 x) + 5 x^{2} = f (2 x) + x^{2} + 1 . . . (i i)$ $(i) + (i i) :$ $2 f (x) + f (2 x) + f (3 x) + 5 x^{2} = f (3 x) + x^{2} + 1 + f (2 x) + x^{2} + 1$ $2 f (x) = - 3 x^{2} + 2$ $\Rightarrow f (x) = - \frac{3}{2} x^{2} + 1$ $\Rightarrow f (10) = - \frac{3}{2} \times 10^{2} + 1 = - 149$
	Answered by Rasheed.Sindhi last updated on 22/Aug/20

	$y = 2 x :$ $f (x) + f (2 x + 2 x) + 5 x (2 x) = f (3 x - 2 x) + x^{2} + 1$ $f (x) + f (4 x) + 10 x^{2} = f (x) + x^{2} + 1$ $f (4 x) = - 9 x^{2} + 1 . . . . . . . . . . . . . . . B$ $L e t 4 x = 10 \Rightarrow x = 5 / 2$ $f (10) = - 9 {(\frac{5}{2})}^{2} + 1$ $= \frac{- 225 + 4}{4} = - \frac{221}{4}$ $W h y i s m y a n s w e r d i f f e r e n t$ $f r o m t h a t o f m r W s i r ?$
	Commented by mr W last updated on 22/Aug/20

	$i t m e a n s t h a t$ $f (x) + f (2 x + y) + 5 x y = f (3 x - y) + x^{2} + 1$ $i s n o t c o n s i s t e n t b e c a u s e i t l e a d s t o$ $c o n t r a d i c t i o n .$
	Commented by Rasheed.Sindhi last updated on 22/Aug/20

	$S i r h o w c a n t h e y b e$ $d i f f e r e n t i a t e d f r o m t h e$ $^{'} c o s i s t e n t f u n c t i o n a l {e q u a t i o n s}^{'} ?$
	Commented by mr W last updated on 22/Aug/20

	$i {c a n}^{'} t e x a c t l y e x p l a i n w h e r e t h e$ $e r r o r l i e s .$
	Commented by Rasheed.Sindhi last updated on 22/Aug/20

	$T han k s sir!$
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