In Calculating the expectation value of the quantum harmonic oscillator, I've come across a problem for finding $\left \langle x \right \rangle$ for the coherent state $\left| \alpha \right \rangle$
$$x = \sqrt{\frac{\hbar}{2m\omega}}(a_+ + a_- )$$
$$\sqrt{\frac{\hbar}{2m\omega}}\left \langle \alpha \right| (a_+ + a_- ) \left| \alpha \right \rangle =\sqrt{\frac{\hbar}{2m\omega}} \alpha ^* (a_+ \alpha + a_- \alpha )$$
This doesn't agree with the result my book has given, which is $\sqrt{\frac{\hbar}{2m\omega}} (\alpha + \alpha ^* )$.
I'm not used to working in bra-ket notation, should it's extremely likely that I'm missing something small here. The general principle is that $\left \langle \beta \right| A \left| \beta \right \rangle = \beta ^* (A\beta)$, right?