Answer
$P(x)=x^2-x+3$
Work Step by Step
Let $P(x)=ax^2+bx+c$ satisfy $P(2)=5$, $P'(2)=3$, and $P''(2)=2$.
Find $P'(x)$ and $P''(x)$:
$P'(x)=2ax+b$
$P''(x)=2a$
Since $P''(2)=2$ and $P''(x)=2a$, we get
$2a=2$
$a=1$
So,
$P'(x)=2\cdot 1x+b$
$P'(x)=2x+b$
Since $P'(2)=3$ and $P'(x)=2x+b$, we get
$2\cdot 2+b=3$
$4+b=3$
$b=-1$
So,
$P(x)=1x^2+(-1)x+c$
$P'(x)=x^2-x+c$
Since $P(2)=5$ and $P'(x)=x^2-x+c$, we get
$2^2-2+c=5$
$4-2+c=5$
$2+c=5$
$c=3$
Thus, $P(x)=x^2-x+3$.
