2重根号

\(0\leq a\pm|b|\sqrt{c}\)のとき、

\[ \sqrt{a\pm|b|\sqrt{c}}=\frac{\sqrt{2}}{2}\left(\sqrt{a+\sqrt{a^{2}-b^{2}c}}\pm\sqrt{a-\sqrt{a^{2}-b^{2}c}}\right) \]

\(a^{2}-b^{2}c\)が平方数のとき2重根号が外せる

\[ \alpha_{\pm}=\sqrt{a\pm|b|\sqrt{c}} \]

とおくと、

\begin{align*} \alpha_{\pm}{}^{2}+\alpha_{\mp}{}^{2} & =a\pm|b|\sqrt{c}+a\mp|b|\sqrt{c}\\ & =2a \end{align*}

\begin{align*} \alpha_{\pm}\alpha_{\mp} & =\sqrt{\left(a\pm|b|\sqrt{c}\right)\left(a\mp|b|\sqrt{c}\right)}\\ & =\sqrt{a^{2}-b^{2}c} \end{align*}

より、

\begin{align*} \alpha_{\pm} & =\frac{\left(\alpha_{\pm}+\alpha_{\mp}\right)+\left(\alpha_{\pm}-\alpha_{\mp}\right)}{2}\\ & =\frac{\left|\alpha_{\pm}+\alpha_{\mp}\right|\pm\left|\alpha_{\pm}-\alpha_{\mp}\right|}{2}\\ & =\frac{\sqrt{\left(\alpha_{\pm}+\alpha_{\mp}\right)^{2}}\pm\sqrt{\left(\alpha_{\pm}-\alpha_{\mp}\right)^{2}}}{2}\\ & =\frac{\sqrt{\alpha_{\pm}{}^{2}+\alpha_{\mp}{}^{2}+2\alpha_{\pm}\alpha_{\mp}}\pm\sqrt{\alpha_{\pm}{}^{2}+\alpha_{\mp}{}^{2}-2\alpha_{\pm}\alpha_{\mp}}}{2}\\ & =\frac{\sqrt{2}}{2}\left(\sqrt{a+\sqrt{a^{2}-b^{2}c}}\pm\sqrt{a-\sqrt{a^{2}-b^{2}c}}\right) \end{align*}

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2重根号

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