Use the quadratic formula to solve the equation. All solutions are nonreal complex numbers. \( x^{2}-x+12=0 \) The solution set is \( \} \).

To solve the equation \( x^{2}-x+12=0 \) using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), we first identify \( a = 1 \), \( b = -1 \), and \( c = 12 \). Plugging these values into the discriminant, we get: \[ b^2 - 4ac = (-1)^2 - 4(1)(12) = 1 - 48 = -47. \] Since the discriminant is negative, the solutions will be nonreal complex numbers. Now we can plug this into the quadratic formula: \[ x = \frac{-(-1) \pm \sqrt{-47}}{2(1)} = \frac{1 \pm i\sqrt{47}}{2}. \] Thus, the solution set is \( \left\{ \frac{1 + i\sqrt{47}}{2}, \frac{1 - i\sqrt{47}}{2} \right\} \). Now, moving on to some fun facts about complex numbers! They may seem tricky, but they are widely used. Engineers apply them in electrical engineering to analyze circuitry, transforming waves and signals. Plus, they have their own unique arithmetic rules that blend beautifully with geometry, creating fascinating patterns in visual math! Want to dive deeper into the world of complex numbers? One engaging area is fractals. Known for their stunning visuals, fractals are created using complex numbers and can be explored through the famous Mandelbrot set. Grab a graphing calculator or a computer program, and get ready to embark on a colorful journey in math that shows that equations can lead to art!