29. \( x^{2}+4=0 \)

To solve the equation \( x^{2}+4=0 \), we first isolate the \( x^2 \) term by subtracting 4 from both sides, giving us \( x^2 = -4 \). Since we can’t have a negative number under a square root in the realm of real numbers, we utilize imaginary numbers. Thus, taking the square root, we find \( x = \pm 2i \), where \( i \) is the imaginary unit. Voilà! The solutions are \( 2i \) and \( -2i \). You might be wondering where this imaginary unit comes from. Historically, complex numbers were introduced in the 16th century when mathematicians were grappling with equations that didn’t have real solutions. They needed a way to handle square roots of negative numbers, leading to the birth of imaginary numbers, and eventually, the beautiful world of complex analysis your math teacher loves to talk about! In real-world applications, these imaginary numbers aren’t just for theoretical fun! They show up in several fields like engineering, quantum physics, and signal processing. For example, electrical engineers use complex numbers extensively in alternating current (AC) circuit analysis to represent the phase and amplitude of voltages and currents. So, while \( 2i \) might sound like a fancy math trick, it plays a significant role in the tech we use every day!