31. \( x^{2}=-12 \)

To solve the equation \( x^{2} = -12 \), we can take the square root of both sides. However, since the right side is negative, we introduce imaginary numbers. Therefore, \( x = \pm \sqrt{-12} = \pm \sqrt{12} \cdot i = \pm 2\sqrt{3}i \). So, the solutions are \( x = 2\sqrt{3}i \) and \( x = -2\sqrt{3}i \). Imaginary numbers, which arise from the square roots of negative numbers, have extensive applications in fields like electrical engineering and quantum physics! For instance, in electrical engineering, imaginary numbers help describe alternating current (AC) circuits, allowing engineers to manage phase shifts and impedance calculations efficiently. When dealing with equations that include square roots of negative values, a common mistake is to ignore the presence of imaginary numbers and to assume there are no solutions. Always remember to incorporate \( i \), the imaginary unit, whenever you encounter a negative under a square root in mathematics—it's your key to unlocking a whole new realm of solutions!