Question

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

Answer 1

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

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