\begin{tabular}{l} What are the roots of the equation? \\ \( \qquad-3 x+19=-x^{2}+1 \) \\ Answer \\ \( \begin{array}{l}\frac{3+9 i}{2} \\ \frac{-3+1 \sqrt{63}}{2} \\ \frac{3+i \sqrt{63}}{2}\end{array} \) \\ \hline Answer \end{tabular}

To tackle this equation, let's first rearrange it into standard form. Rewriting \(-3x + 19 = -x^2 + 1\) gives us the quadratic equation \(x^2 - 3x + 18 = 0\). Now, applying the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) with \(a = 1\), \(b = -3\), and \(c = 18\) leads to complex solutions, showcasing the beauty of mathematics where even "imaginary" numbers lend clarity and provide insight into the nature of the roots! Understanding the roots helps not just in solving equations but also in graphing them! This quadratic equation describes a parabola that opens upward and has complex roots, which means its vertex is above the x-axis, reflecting that it doesn't intersect with the line \(y=0\). This is a common scenario in algebra where visualizing the graph can aid in comprehending the behavior of functions, especially when you encounter complex or imaginary numbers!