Question

dada la funcion cuadratica.
f(x) =ax^{2} + bx+ c,si f(x -1) + f(x-2) =5x^{2} - 2x +9,hallar a+b+c

Answer 1

Resolvemos:

f(x - 1) = a (x - 1)² + b (x - 1) + c; quitamos paréntesis:

f(x - 1) = a x² + (b - 2 a) x + a - b + c (*)

f(x - 2) = a (x - 2)² + b (x - 2) + c; quitamos paréntesis:

f(x - 2) = a x² + (b - 4 a) x + 4 a - b + c (**)

Sumamos (*) + (**):

2 a x² + 2 (b - 3 a) x + 5 a - 3 b + 2 c

Identificamos coeficientes.

2 a = 5; a = 5/2

2 (b - 3 . 5/2) = - 2; b - 15/2 = - 1; b = 13/2

5 . 5/2 - 3 . 13/2 + 2 c = 9

c = 1/2 (9 + 39/2 - 25/2); c = 8

a + b + c = 5/2 + 13/2 + 8 = 17

Mateo