[tex]\sqrt{4-\sqrt 7}= \sqrt a - \sqrt b \\ \\ \sqrt{4-\sqrt 7}^2=a+b-2\sqrt{ab}\\ \\ 4-\sqrt 7 = a+b-\sqrt{4ab}\\ \\ a+b = 4 \\ 4ab = 7\\ \\ a+b = 4 \\ ab =\dfrac{7}{4} \\ \\ t^2-4t+\dfrac{7}{4} = 0 \\ \\\Delta = 16-7 = 9 \Rightarrow t_{1,2} = \dfrac{4\pm 3}{2} \\ \\ \Rightarrow a = \dfrac{7}{2},b = \dfrac{1}{2} \\ \\ \Rightarrow \sqrt{4-\sqrt 7} = \sqrt{\dfrac{7}{2}}-\sqrt{\dfrac{1}{2}}[/tex]
[tex]\Rightarrow \sqrt{4-\sqrt 7} = \dfrac{\sqrt{14}}{2} - \dfrac{\sqrt 2}{2}\\ \\ \text{Analog: } \sqrt{4+\sqrt 7} = \dfrac{\sqrt{14}}{2} + \dfrac{\sqrt 2}{2}[/tex]
