Question

известно, что х1, х2- корни уравнения
${2x}^{2} - ( \sqrt{3} + 5)x - \sqrt{4 + 2 \sqrt{3} } = 0$

Answer 1

$2x^2-(\sqrt3+5)x-\sqrt{4+2\sqrt3}=0\\\\\star \; \sqrt{4+2\sqrt3}=\sqrt{1+3+2\cdot 1\cdot \sqrt3}=\sqrt{(1+\sqrt3)^2}=1+\sqrt3\; \star \\\\2x^2-(\sqrt3+5)x-(1+\sqrt3)=0\\\\D=(\sqrt3+5)^2+8\cdot (1+\sqrt3)=28+10\sqrt3+8+8\sqrt3=36+18\sqrt3=\\\\=9(4+2\sqrt3)=3^2\cdot (1+\sqrt3)^2$

$x_1=\frac{\sqrt3+5-3(1+\sqrt3)}{4}=\frac{2-2\sqrt3}{4}=\frac{1-\sqrt3}{2}\; ,\\\\x_2=\frac{\sqrt3+5+3(1+\sqrt3)}{4}=\frac{8+4\sqrt3}{4}=2+\sqrt3$