f-x-ax-2-bx-c-f-x-1-f-x-f-x-1-x-2-1-f-2-

Question Number 192083 by sciencestudentW last updated on 07/May/23

f (x) = {a x}^{2} + b x + c

f (x - 1) + f (x) + f (x + 1) = x^{2} + 1

f (2) = ?

Answered by AST last updated on 07/May/23

L i n e 2

\Rightarrow a ({(x - 1)}^{2} + x^{2} + {(x + 1)}^{2}) + b (x - 1 + x + x + 1) + 3 c

= x^{2} + 1 \Rightarrow a (3 x^{2} + 2) + b (3 x) + 3 c = x^{2} + 1

\Rightarrow 3 {a x}^{2} + 3 b x + 2 a + 3 c = x^{2} + 1

\Rightarrow 3 a = 1 \Rightarrow a = \frac{1}{3}; b = 0; 2 a + 3 c = 1 \Rightarrow c = \frac{1}{9}

\Rightarrow f (x) = \frac{1}{3} x^{2} + \frac{1}{9} = \frac{3 x^{2} + 1}{9}

\Rightarrow f (2) = \frac{13}{9}

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