Question

DAU FUNDA!! Cat face :
$\sqrt{5 + \sqrt{5} }$

Answer 1

Nu se poate rezolva, decat daca ii calculezi valoarea aproximata, aceasta fiind 2,68999

Answer 2

[tex]\sqrt{5+\sqrt5} =\sqrt{a}+\sqrt{b}\Big|^2\\ 5+\sqrt5 = a+b+2\sqrt{ab} \\5+\sqrt5 = a+b+\sqrt{4ab} \\ \\ \left\{ \begin{array}{ll} a+b = 5 \\ 4ab = 5 \end{array} \right \Rightarrow \left\{ \begin{array}{ll} a+b = 5 \\ ab = \dfrac{4}{5} \end{array} \right | \Rightarrow t^2-St+P = 0 \Rightarrow \\ \\ \Rightarrow t^2-5t+\dfrac{4}{5}= 0 \Big|\cdot 5 \Rightarrow 5t^2-25t+4 = 0 \\ \\ \Delta = 25^2-4\cdot 5\cdot 4 = 625 - 80 = 545 \\ \\t_{1,2} = \dfrac{25\pm\sqrt{545}}{10} \\ \\ [/tex]

$\Rightarrow \boxed{\boxed{ \sqrt{5+\sqrt{5}} = \sqrt{\dfrac{25+\sqrt{545}}{10}}+\sqrt{\dfrac{25-\sqrt{545}}{10}}}}$