Rozwiąż równanie
: 30 sty 2012, 10:30
\(2^{x+ \sqrt{x^2-4}}-5* \sqrt{2}^{x-2+ \sqrt{x^2-4} }-6=0\)
\(2^{x+\sqrt{x^2-4}}-5\cdot(\sqrt{2})^{x-2+\sqrt{x^2-4}}-6=0\)
\(x^2-4\ge0\\x\in(-\infty;\ -2>\ \cup\ <2;\ \infty)\)
\(2^{x+\sqrt{x^2-4}}-5\cdot\frac{1}{2}\cdot2^{\frac{1}{2}(x+\sqrt{x^2-4})}-6=0\\2^{\frac{1}{2}(x+\sqrt{x^2-4})}=t>0\\t^2-2,5t-6=0\\2t^2-5t-12=0\\\Delta=25+96=121\\t_1=\frac{5-11}{4}<0\ \vee\ t_2=\frac{5+11}{4}=4\\t>0\\t=4\)
\(2^{\frac{1}{2}(x+\sqrt{x^2-4})}=4\\\frac{1}{2}(x+\sqrt{x^2-4})=2\\x+\sqrt{x^2-4}=4\\\sqrt{x^2-4}=4-x\\\sqrt{x^2-4}\ge0\\4-x\ge0\\x\le4\\x^2-4=16-8x+x^2\\8x=20\\x=2,5\)