First, recall that given the equation
[tex]ax^2+b+c=0[/tex]
The solutions are given by
[tex]x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a}[/tex]
where the sign in the middle means that we get one root by taking a plus sign and we get the other root by taking a minus sign.
In our case, we have a=1, b=-6 and c=-41. So the solutions of this equation are given by
[tex]x=\frac{-(-6)\pm\sqrt[]{(-6)^2-4(-41)}}{2}=\frac{6\pm\sqrt[]{36+164}}{2}=\frac{6\pm\sqrt[]{200}}{2}[/tex]
Note that
[tex]\sqrt[]{200}=\sqrt[]{100\cdot2}=\sqrt[]{100}\cdot\sqrt[]{2}=10\sqrt[]{2}[/tex]
Then
[tex]x=\frac{6\pm10\sqrt[]{2}}{2}=3\pm5\sqrt[]{2}[/tex]
taking sqrt(2) as 1.4142, we get
[tex]x=3+5\cdot\sqrt[]{2}=10.071\approx10[/tex]
and
[tex]x=3-5\sqrt[]{2}=-4.071\approx-4[/tex]
