11) \( k^{2}-4 k+1=-5 \)

To solve the equation \( k^{2} - 4k + 1 = -5 \), first, let's simplify it by adding 5 to both sides, resulting in \( k^{2} - 4k + 6 = 0 \). Now, we can use the quadratic formula \( k = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Here, \( a = 1 \), \( b = -4 \), and \( c = 6 \). Plugging in these values, we find \( k = \frac{4 \pm \sqrt{(-4)^2 - 4 \cdot 1 \cdot 6}}{2 \cdot 1} = \frac{4 \pm \sqrt{16 - 24}}{2} = \frac{4 \pm \sqrt{-8}}{2} \). This gives us \( k = \frac{4 \pm 2i\sqrt{2}}{2} = 2 \pm i\sqrt{2} \). Therefore, the solutions are \( k = 2 + i\sqrt{2} \) and \( k = 2 - i\sqrt{2} \). The equation quite clearly establishes that \( k \) has complex solutions, indicating there's a rich mathematical landscape behind quadratic functions. In real-world applications, such as electrical engineering and signal processing, complex numbers can represent waveforms and oscillations, illustrating how abstract math finds practical relevance in technology and design!