Re-writing in sign basis. $\newcommand\ket[1]{\left\vert #1\right\rangle}$ Let $\ket\phi = 12 \ket{0} + 1 + 2\sqrt{i2}\ket{1}$. Write $\ket\phi$ in the form $\alpha_0\ket{+} + \alpha_1\ket{-}$. What is $\alpha

Re-writing in sign basis. $\newcommand\ket[1]{\left\vert #1\right\rangle}$ Let $\ket\phi = 12 \ket{0} + 1 + 2\sqrt{i2}\ket{1}$. Write $\ket\phi$ in the form $\alpha_0\ket{+} + \alpha_1\ket{-}$. What is $\alpha_0$? I came across this problem in a course i am doing, i have been struggling writing things in sign basis, much appreciated.

The "sign basis" is defined as $$\begin{aligned} \lvert+\rangle &= \frac{1}{\sqrt{2}}\left(\lvert0\rangle+\lvert1\rangle\right)\\\ \lvert-\rangle &= \frac{1}{\sqrt{2}}\left(\lvert0\rangle-\lvert1\rangle\right) \end{aligned}$$ Those equations can be solved for $\lvert0\rangle$ and $\lvert1\rangle$. Then it's just a matter of inserting in the given state and comparing the coefficients.