2f(x−1) +3f(x+1) ^ = 3x^2 −5x find f(x)


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Question Number 82701 by jagoll last updated on 23/Feb/20

$2 f (x - 1) + 3 f (x + 1) \overset{}{} = 3 x^{2} - 5 x$ $f i n d f (x)$
	Answered by TANMAY PANACEA last updated on 23/Feb/20

	$f (x) = {a x}^{2} + b x + c$ $2 f (x - 1) + 3 f (x + 1)$ $2 [a {(x - 1)}^{2} + b (x - 1) + c] + 3 [a {(x + 1)}^{2} + b (x + 1) + c]$ $= [2 {a x}^{2} - 4 a x + 2 a + 2 b x - 2 b + 2 c + 3 {a x}^{2} + 6 a x + 3 a + 3 b x + 3 b + 3 c]$ $= x^{2} (5 a) + x (2 a + 5 b) + (2 a - 2 b + 2 c + 3 a + 3 b + 3 c)$ $5 a = 3 \to a = \frac{3}{5} 2 a + 5 b = - 5$ $2 \times \frac{3}{5} + 5 b = - 5$ $5 b = \frac{- 31}{5} \to b = \frac{- 31}{25}$ $5 a + b + 5 c = 0$ $5 \times \frac{3}{5} + \frac{- 31}{25} + 5 c = 0$ $75 - 31 + 125 c = 0 \to c = \frac{- 44}{125}$ $f (x) = \frac{3}{5} x^{2} - \frac{31}{25} x - \frac{44}{125} p l s c h e c k$
	Commented by jagoll last updated on 23/Feb/20

	$o o o k s i r . t h a n k s m u c h$
	Commented by TANMAY PANACEA last updated on 23/Feb/20

	$i h a v e a s s u m e d b e c a u s e r i g h t h a n d s i d e s h o w$ $x^{2}$
	Commented by jagoll last updated on 23/Feb/20

	$s i r, w h e r e d o w e s u p p o s e t h a t$ $t h e f u n c t i o n f (x) i s q u a d r a t i c$ $f u n c t i o n ?$
	Commented by mr W last updated on 23/Feb/20

	$t o j a g o l l s i r :$ $y o u m a y a s s u m e f (x) = \underset{k = 0}{\sum^{\infty}} a_{k} x^{k}$ $b u t a s s o o n a s A f (x - 1) + B f (x + 1) i s$ $p o l y n o n m i a l s e c o n d o r d e r, y o u w i l l$ $a u t o m a t i c a l l y g e t a_{k} = 0 f o r k ⩾ 3 .$
	Commented by mr W last updated on 23/Feb/20

	$i t i s t h e s a m e m e t h o d t o u s e f o r s o l v i n g$ $q u e s t i o n s l i k e :$ $i f a_{1} = 1 a n d$ $3 a_{n + 1} + 2 a_{n - 1} = 3 n^{2} - 5 n$ $f i n d a_{n} = ?$
	Commented by jagoll last updated on 23/Feb/20

	$o o y e s . i r e m e m b e r t h e m e t h o d$ $t h a t y o u e x p l a i n e d a f e w d a y s a g o$
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