Given the equation:
[tex]L=\frac{1}{\sqrt{8-2x-x^2}}[/tex]
Let's solve the equation for x.
To solve for x, the first step is to cross multiply:
[tex]L*\sqrt{8-2x-x^2}=1[/tex]
Now, divide both sides by L:
[tex]\begin{gathered} \frac{L*\sqrt{8-2x-x^2}}{L}=\frac{1}{L} \\ \\ \sqrt{8-2x-x^2}=\frac{1}{L} \end{gathered}[/tex]
Square both sides:
[tex]\begin{gathered} (\sqrt{8-2x-x^2})\text{ }^2=\text{ \lparen}\frac{1}{L})^2 \\ \\ 8-2x-x^2=\frac{1}{L^2} \\ \\ 8-2x-x^2=L^{-2} \end{gathered}[/tex]
Now, subtract L⁻² from both sides and equate to zero:
[tex]\begin{gathered} 8-2x-x^2-L^{-2}=L^{-2}-L^{-2} \\ \\ 8-2x-x^2-L^{-2}=0 \\ \\ \text{ Rearrange:} \\ -x^2-2x+8-L^{-2}=0 \end{gathered}[/tex]
Solve the equation using the quadratic formula:
[tex]x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}[/tex]
Where:
a = -1
b = -2
c = (8 - L⁻²)
Thus, we have:
[tex]\begin{gathered} x=\frac{-(-2)\pm\sqrt{-2^2-4(-1)(8-L^{-2})}}{2(-1)} \\ \\ x=\frac{2\pm\sqrt{4+4(8-L^{-2})}}{-2} \end{gathered}[/tex]
Therefore, the solution is:
[tex][/tex]
