The number of real solutions of the equation $2^{x/2}+(\sqrt{2}+1)^x=(5+2\sqrt{2})^{x/2}$, is |
1 2 4 infinite |
1 |
We have, $2^{x/2}+(\sqrt{2}+1)^x=(5+2\sqrt{2})^{x/2}$ $(\sqrt{2})^x+(\sqrt{2}+1)^x=\{\sqrt{(\sqrt{2})^2+(\sqrt{2}+1)^2}\}^x$ $⇒(\sqrt{2})^x+(\sqrt{2}+1)^x=(\sqrt{5+2\sqrt{2}})^x$ $⇒(\frac{\sqrt{2}}{\sqrt{5+2\sqrt{2}}})^x+(\frac{\sqrt{2}+1}{\sqrt{5+2\sqrt{2}}})^x=1$ $⇒(\frac{\sqrt{2}}{\sqrt{5+2\sqrt{2}}})^x+(\frac{\sqrt{2}+1}{\sqrt{5+2\sqrt{2}}})^x=(\frac{\sqrt{2}}{\sqrt{5+2\sqrt{2}}})^2+(\frac{\sqrt{2}+1}{\sqrt{5+2\sqrt{2}}})^2$ $⇒x=2$ Therefore, there is only one real solution possible. |