Field extensions of $\mathbb{C}$ of degree 2

Suppose that $E$ is a field extension of degree $2$ of $\mathbb{C}$, and let $x$ be an element of $E\backslash\mathbb{C}$. By definition of degree, we have that the family $(1,x)$ forms a basis of $E$ (viewed as a vector space over $\mathbb{C}$). In particular, this means that $x^{2}$ can be written as a linear combination:
$$x^{2}=ax+b$$where $a$ and $b$ are complex numbers. But any root of a complex polynomial, here $X^2-aX-b$ is in $\mathbb{C}$. This means that $x$ is still in $\mathbb{C}$, which is absurd.