Completing the square of $x^2 - mx = 1$ is not giving me the right answer. This is my attempt
$$
\begin{align}
    x^2 - mx &= 1 \\
    x^2 - mx - 1 &= 0 \\
    \left(x^2 - mx + \frac{m^2}{4} - \frac{m^2}{4}\right) - 1 &= 0 \\
    \left(x^2 - mx + \frac{m^2}{4}\right) - \frac{m^2}{4} - 1 &= 0 \\
    \left(x^2 - mx + \frac{m^2}{4}\right) - \frac{m^2}{4} - \frac{4}{4} &= 0 \\
    \left(x^2 - mx + \frac{m^2}{4}\right) - \frac{m^2 - 4}{4} &= 0 \\
    \left(x^2 - mx + \frac{m^2}{4}\right) &= \frac{m^2 - 4}{4} \\
    \left(x - \frac{m}{2}\right)^2 &= \frac{m^2 - 4}{4} \\
    \sqrt{\left(x - \frac{m}{2}\right)^2} &= \sqrt{\frac{m^2 - 4}{4}} \\
    x - \frac{m}{2} &= \pm \frac{\sqrt{m^2 - 4}}{\sqrt{4}} \\
    x &= \frac{m}{2} \pm \frac{\sqrt{m^2 - 4}}{2} \\[20pt]
    x_1 &= \frac{m}{2} - \frac{\sqrt{m^2 - 4}}{2} \\
    x_1 &= \frac{m - \sqrt{m^2 - 4}}{2} \\[16pt]
    x_2 &= \frac{m}{2} + \frac{\sqrt{m^2 - 4}}{2} \\
    x_2 &= \frac{m + \sqrt{m^2 - 4}}{2} \\
\end{align}
$$
However, the correct answer according to the text is:
$$
\begin{align}
x_1 &= \frac{m}{2} - \frac{\sqrt{m^2 + 4}}{2} \\
x_2 &= \frac{m + \sqrt{m^2 + 4}}{2} \\
\end{align}
$$
Why $\sqrt{m^2 + 4}$ instead of $\sqrt{m^2 - 4}$ ???

When you combined $$-\frac{m^2}{4} - \frac{4}{4}$$ into one fraction, you wrote it as $$-\frac{m^2-4}{4}.$$ You should have gotten $$-\frac{m^2+4}{4}$$ to make both terms appropriately negative.
