1 Answer

Johnwards · Answer 1 · 2023-01-07T03:34:56+0000

p(x-1) is a quadratic in the form

a(x-1)^2 + b(x-1) + c

We want to write
p(x-1) instead in the form

a' x^2 + b' x + c

To do this, simply expand the first form completely.

$a(x-1)^2 + b(x-1) + c = a(x^2 - 2x + 1) + b(x-1) + c \\\\ ~~~~~~~~ = ax^2 + (-2a + b)x + (a - b + c)$

Then
a'=a ,
b'=-2a+b , and
c'=a-b+c .

Given

p(x-1) = 2(x-1)^2 + 3(x-1) + 4

(so
a=2 ,
b=3 , and
c=4 ) it follows that

a' = a = 2

b' = -2a + b = -2^2 + 3 = -1

c' = a - b + c = 2 - 3 + 4 = 3

$\implies p(x-1) = \boxed{2}\,x^2 - \boxed{1}\,x + \boxed{3}$

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