Find all solutions of the equation $\cos 2θ + 7 \cos θ = 8$ Find all solutions of the equation in the interval [0, 2π) (express your answer in terms of $k$, where $k$ is any integer).
$$\cos 2\theta + 7 \cos\theta = 8$$ 
Did I make a mistake somewhere? My final answer was $0+2\pi k$
Here is my work. I went from step 1 to step 2 by using the trig identity $\cos 2\theta = 2\cos^2\theta -1$ and then solved it like a quadratic. I also set $\cos \theta = x$ because it's easier for me to visualize the equation that way.


Notice, we have $$\cos 2\theta+7\cos \theta=8$$ $$2\cos^2\theta-1+7\cos \theta=8$$ $$2\cos^2\theta+7\cos \theta-9=0$$ $$\underbrace{2\cos^2\theta+9\cos \theta}\underbrace{-2\cos \theta-9}=0$$
$$\cos\theta(2\cos \theta+9)-(2\cos \theta+9)=0$$ $$(\cos \theta-1)(2\cos \theta+9)=0$$ $$\implies \cos\theta-1=0\iff \cos\theta=1=\cos 0$$ Writing the general solution, we get $$\theta=2k\pi\pm 0=2k\pi$$
$$\implies 2\cos\theta+9=0\implies  \cos\theta \neq \frac{-9}{2}$$ this case does not give any solution.
Hence, the general solution of given equation: $\cos 2\theta+7\cos \theta=8$ is $$\bbox[5px, border:2px solid #C0A000]{\color{red}{\theta=2k\pi}}$$
Where, $\color{red}{k}$ is any integer.
