Step by step explanation:
Recall that the quadratic formula states that the solutions of the quadratic equation
[tex]ax^2+bx+c=0[/tex]
are:
[tex]\frac{-b\pm\sqrt[]{b^2-4ac}}{2a}.[/tex]
Now, to use the quadratic formula we have to take the given equation to ax²+bx+c=0 form:
[tex]\begin{gathered} 11x^2-4x-1=1-1, \\ 11x^2-4x-1=0. \end{gathered}[/tex]
Therefore a=11, b=-4, and c=-1.
Substituting the above values in the quadratic formula we get:
[tex]\frac{-(-4)\pm\sqrt[]{(-4)^2-4(11)(-1)}}{2(11)}\text{.}[/tex]
Simplifying the above expression we get:
[tex]\begin{gathered} \frac{4\pm\sqrt[]{16+44}}{22}=\frac{4\pm\sqrt[]{60}}{22}=\frac{4\pm\sqrt[]{4\cdot15}}{22} \\ =\frac{4\pm\sqrt[]{4}\sqrt[]{15}}{22}=\frac{4\pm2\sqrt[]{15}}{22} \\ =\frac{2\cdot2\pm2\sqrt[]{15}}{2\cdot11}=\frac{2\pm\sqrt[]{15}}{11}=\frac{2}{11}\pm\frac{\sqrt[]{15}}{11}\text{.} \end{gathered}[/tex]
Answer: First option.
