Let \(E/\QQ\) be a field extension with \([E:\QQ] = 2\). Then \(E = \QQ(\sqrt q)\) for some squarefree integer \(q \notin \theset{0, 1}\).
Choose \(\alpha\in E\setminus\QQ\). Since \(\QQ \subsetneq \QQ(\alpha) \subseteq E\) and \([E:\QQ]=2\), we have \(E = \QQ(\alpha)\), and the minimal polynomial of \(\alpha\) over \(\QQ\) has the form \(x^2 + bx + c\) with \(b, c\in\QQ\). By the quadratic formula, \(\alpha = \frac{-b \pm \sqrt d}{2}\) with \(d \coloneqq b^2 - 4c\in\QQ\), so \(E = \QQ(\sqrt d)\), and \(d\) is not a square in \(\QQ\) because \(\alpha\notin\QQ\). Write \(d = m/n\) with \(m, n\in\ZZ\) and \(n \neq 0\); then \(\sqrt d = \sqrt{mn}/n\), so \(E = \QQ(\sqrt{mn})\). Write \(mn = s^2 q\) with \(s\in\ZZ\setminus\theset 0\) and \(q\) a squarefree integer; then \(E = \QQ(\sqrt q)\), and \(q\notin\theset{0,1}\) because \(d\) is not a square in \(\QQ\).