\(E=\sqrt{a+3+4\sqrt{a-1}}+\sqrt{a+3-4\sqrt{a-1}}\)(Đkxd: a≥1)
\(E^2=\left(\sqrt{a+3+4\sqrt{a-1}}+\sqrt{a+3-4\sqrt{a-1}}\right)^2\)
\(E^2=a+3+4\sqrt{a-1}+2\sqrt{\left(a+3+4\sqrt{a-1}\right)\left(a+3-4\sqrt{a-1}\right)}+a+3-4\sqrt{a-1}\)
\(E^2=2a+6+2\sqrt{\left[\left(a+3\right)^2-16\left(a-1\right)\right]}\)
\(E^2=2a+6+2\sqrt{a^2-10a+25}\)
\(E^2=2a+6+2\sqrt{\left(a-5\right)^2}\)
\(E^2=2a+6+2\left|a-5\right|\)
Th1 a≥5 ta có
\(E^2=2a+6+2\text{}\left(a-5\right)=4a-4\)
\(E=\sqrt{4a-4}=2\sqrt{a-1}\)
Th2 a<5 ta có
\(E^2=2a+6-2\text{}\left(a-5\right)=16\)
E=4
Ta có: \(E=\sqrt{a+3+4\sqrt{a-1}}+\sqrt{a+3-4\sqrt{a-1}}\)
\(=\sqrt{a-1+2\cdot\sqrt{a-1}\cdot2+4}+\sqrt{a-1-2\cdot\sqrt{a-1}\cdot2+4}\)
\(=\left|\sqrt{a-1}+2\right|+\left|\sqrt{a-1}-2\right|\)
