Question

упростить выражение и подставить X​

Answer 1

Ответ:

a-b

Пошаговое объяснение:

$\frac{2b\sqrt{x^{2} -1} }{x-\sqrt{x^{2} -1}} =\frac{(2b\sqrt{x^{2} -1})*(x+\sqrt{x^{2} -1}) }{(x-\sqrt{x^{2} -1})*(x+\sqrt{x^{2} -1})} =\frac{2bx\sqrt{x^{2} -1}+2b(\sqrt{x^{2} -1})^{2} }{x^{2} -(x^{2} -1)} =\\=\frac{2bx\sqrt{x^{2} -1}+2b(x^{2} -1)}{x^{2} -(x^{2} -1)} =\frac{2bx\sqrt{x^{2} -1}+2b(x^{2} -1)}{x^{2} -x^{2} +1} =\\=\frac{2bx\sqrt{x^{2} -1}+2b(x^{2} -1)}{1} =2bx\sqrt{x^{2} -1}+2b(x^{2} -1)$

$x=\frac{1}{2} (\sqrt{\frac{a}{b} } +\sqrt{\frac{b}{a} } )=\frac{1}{2} \frac{(\sqrt{a} )^{2}+(\sqrt{b} )^{2}}{\sqrt{ab} } =\frac{a+b}{2\sqrt{ab} }$

$2b(\frac{a+b}{2\sqrt{ab} })\sqrt{(\frac{a+b}{2\sqrt{ab} })^{2} -1}+2b((\frac{a+b}{2\sqrt{ab} })^{2} -1)=\\=b(\frac{a+b}{\sqrt{ab} })\sqrt{\frac{(a+b)^{2}}{4ab } -1}+2b(\frac{(a+b)^{2}}{4ab } -1)=\\=b(\frac{a+b}{\sqrt{ab} })\sqrt{\frac{a^{2}+2ab+b^{2}-4ab}{4ab } }+(\frac{a^{2}+2ab+b^{2}-4ab}{2a } )=\\=b(\frac{a+b}{\sqrt{ab} })\sqrt{\frac{a^{2}-2ab+b^{2}}{4ab } }+\frac{a^{2}-2ab+b^{2}}{2a }=$

$=b\frac{(a+b)}{\sqrt{ab} } \frac{\sqrt{a^{2}-2ab+b^{2}} }{2\sqrt{ab} }  +\frac{a^{2}-2ab+b^{2}}{2a }=b\frac{(a+b)}{2ab }\sqrt{(a-b)^{2}}  +\frac{a^{2}-2ab+b^{2}}{2a }=\\=\frac{(a+b)(a-b)}{2a }+\frac{a^{2}-2ab+b^{2}}{2a }=\frac{a^{2}-b^{2}}{2a }+\frac{a^{2}-2ab+b^{2}}{2a }=\\=\frac{a^{2}-b^{2}+a^{2}-2ab+b^{2}}{2a}=\frac{2a^{2}-2ab}{2a}=\frac{2a(a-b)}{2a}=a-b$