2f-x-1-3f-x-1-3x-2-5x-find-f-x-

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

Facebook Tweet Pin