# 1958 AHSME Problems/Problem 46

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## Problem

For values of $x$ less than $1$ but greater than $-4$, the expression $\frac{x^2 - 2x + 2}{2x - 2}$ has:

$\textbf{(A)}\ \text{no maximum or minimum value}\qquad \\ \textbf{(B)}\ \text{a minimum value of }{+1}\qquad \\ \textbf{(C)}\ \text{a maximum value of }{+1}\qquad \\ \textbf{(D)}\ \text{a minimum value of }{-1}\qquad \\ \textbf{(E)}\ \text{a maximum value of }{-1}$

## Solution

From $\frac{x^2 - 2x + 2}{2x - 2}$, we can further factor $\frac{x^2 - 2x + 2}{2(x - 1)}$ and then $\frac{(x-1)^{2}+1}{2(x - 1)}$ and finally $\frac{x-1}{2}+\frac{1}{2x-2}$. Using $AM-GM$, we can see that $\frac{x-1}{2}=\frac{1}{2x-2}$. From there, we can get that $2=2 \cdot (x-1)^{2}$.

From there, we get that $x$ is either $2$ or $0$. Substituting both of them in, you get that if $x=2$, then the value is $1$. If you plug in the value of $x=0$, you get the value of $-1$. So the answer is $\textbf{(E)}$

Solution by: the_referee