[tex]1) x^{2} -7x+6=0\\ a=1 \\b=-7\\c=6\\
\Delta=(-7)^{2}-4*1*6=49-24=25\\ \sqrt{\Delta} = \sqrt{25}=5 \\x_{1}= \frac{-b- \sqrt{\Delta} }{2a}= \frac{7-5}{2}= \frac{2}{2}=1\\ x_{2}= \frac{-b+ \sqrt{\Delta} }{2a}= \frac{7+5}{2}= \frac{12}{2}=6\\
S=[1, 6] \\
2) 4 x^{2} -12x+5=0\\ a=4\\b=-12\\c=5\\
\Delta=(-12)^2-4*4*5=144-80=64\\
x_{1}= \frac{-b- \sqrt{\Delta} }{2a}= \frac{12-8}{2*4}= \frac{4}{8}= \frac{1}{2}\\
x_{2}= \frac{-b+ \sqrt{\Delta} }{2a}= \frac{12+8}{2*4}=\frac{20}{8}=\frac{5}{2}\\
[/tex] [tex]S=[ \frac{1}{2} ; \frac{5}{2} ]\\
3) x^{2} -3 \sqrt{3}x+6=0\\ a=1\\b=-3 \sqrt{3}\\c=6\\ \Delta=b^2-4ac=(-3\sqrt{3})^2-4*1*6=27-24=3\\
x_{1}= \frac{-b- \sqrt{\Delta} }{2a}= \frac{3 \sqrt{3}- \sqrt{3}}{2}= \frac{2 \sqrt{3} }{2}= \sqrt{3}\\
x_{2}= \frac{-b+ \sqrt{\Delta} }{2a}=\frac{3 \sqrt{3}+ \sqrt{3}}{2}= \frac{4 \sqrt{3} }{2}=2 \sqrt{3}\\
S=[ \sqrt{3}; 2 \sqrt{3} ] \\[/tex]
[tex]4) x^{2} -4 \sqrt{2}x-6=0\\ a=1\\b=-4\sqrt{2}\\c=-6\\
\Delta=(-4\sqrt{2})^{2}-4*1*(-6)=32+24=56\\
x_{1}= \frac{-b- \sqrt{\Delta} }{2a}= \frac{4\sqrt{2}-\sqrt{56}}{2}= \frac{4\sqrt{2}-\sqrt{4*14}}{2}= \\=\frac{4\sqrt{2}-2 \sqrt{14} }{2}= \frac{2(2\sqrt{2}-\sqrt{14})}{2}= 2\sqrt{2}-\sqrt{14} \\
x_{2}= \frac{-b+ \sqrt{\Delta} }{2a}= \frac{4\sqrt{2}+\sqrt{56}}{2}= \frac{4\sqrt{2}+\sqrt{4*14}}{2}= \\=\frac{4\sqrt{2}+2 \sqrt{14} }{2}= \frac{2(2\sqrt{2}+\sqrt{14})}{2}= 2\sqrt{2}+\sqrt{14} \\
[/tex]
[tex]S=[2\sqrt{2}-\sqrt{14}; 2\sqrt{2}+\sqrt{14}][/tex]
