Denesting $\sqrt{x\pm\sqrt{y}}$

Pub: Sep. 14, 2014 | Wri: a few years ago

\[ \begin{split} & \sqrt{x+\sqrt{y}} \\ & \begin{split} \! = \; & \sqrt{a}+\sqrt{b} \\ \! = \; & \sqrt{(\sqrt{a}+\sqrt{b})^2} \\ \! = \; & \sqrt{a+2\sqrt{ab}+b} \\ \! = \; & \sqrt{a+b+\sqrt{4ab}} \end{split} \\ \end{split} \qquad \begin{split} & \sqrt{x-\sqrt{y}} \\ & \begin{split} \! = \; & \sqrt{a}-\sqrt{b} \\ \! = \; & \sqrt{(\sqrt{a}-\sqrt{b})^2} \\ \! = \; & \sqrt{a-2\sqrt{ab}+b} \\ \! = \; & \sqrt{a+b-\sqrt{4ab}} \end{split} \\ \end{split} \\[48pt] x = a + b \stag{1} \qquad y = 4ab \stag{2} \\[6pt] b = x-a \stag{3} \\[6pt] (3) \longrightarrow (2) \\[6pt] \begin{split} y &= 4a(x-a) \\[3pt] &= 4ax - 4a^2 \end{split} \\[15pt] 4a^2 - 4ax + y = 0 \\[3pt] a^2 - ax + \f{1}{4}y = 0 \\[3pt] a = \f{x \pm \sqrt{x^2-y}}{2} \\[3pt] \begin{split} b &= x-a \\[3pt] &= \f{2x - \b{x \pm \sqrt{x^2-y}}}{2} \\[3pt] &= \f{x \mp \sqrt{x^2-y}}{2} \end{split} \\[42pt] a = \f{x+\sqrt{x^2-y}}{2} \qquad b = \f{x-\sqrt{x^2-y}}{2} \\[12pt] \sqrt{x\pm\sqrt{y}} = \sqrt{\f{x+\sqrt{x^2-y}}{2}} \pm \sqrt{\f{x-\sqrt{x^2-y}}{2}} \]