Question

Answer 1

\(f\left(5-2\sqrt{3}\right)=f\left(2\right)\\ \Leftrightarrow\sqrt{5-2\sqrt{3}-1}+m\left(5-2\sqrt{3}\right)+2=\sqrt{2-1}+2m+2\\ \Leftrightarrow\sqrt{\left(\sqrt{3}-1\right)^2}+5m-2\sqrt{3}m+2=3+2m\\ \Leftrightarrow\sqrt{3}-1+5m-2\sqrt{3}m+2=3+2m\\ \Leftrightarrow3m-2\sqrt{3}m=2-\sqrt{3}\\ \Leftrightarrow m\left(3-2\sqrt{3}\right)=2-\sqrt{3}\\ \Leftrightarrow m=\dfrac{2-\sqrt{3}}{3-2\sqrt{3}}=\dfrac{\left(2-\sqrt{3}\right)\left(3+2\sqrt{3}\right)}{-3}\\ m=\dfrac{\sqrt{3}}{-3}=-\dfrac{\sqrt{3}}{3}\)