Solve this equation and select the choice that contains all of the solutions. \[ 4 x^{2}=-24 \]

To solve the equation \( 4 x^{2} = -24 \), first divide both sides by 4: \[ x^{2} = -6 \] Now we notice that we have a negative value on the right side, indicating that we'll be working with imaginary numbers. Taking the square root of both sides gives us: \[ x = \pm \sqrt{-6} \] This simplifies to: \[ x = \pm i \sqrt{6} \] So the solutions are \( x = i \sqrt{6} \) and \( x = -i \sqrt{6} \). Now, let's explore a bit about imaginary numbers and their real-world applications! Imaginary numbers, like \( i \), are crucial in fields such as engineering and physics, especially when dealing with waveforms and electrical circuits. They help model alternating current, providing insights into phase shifts and resonance that traditional real number solutions can’t offer. Feeling adventurous? Dabbling into the world of complex numbers can open up a treasure trove! A fascinating book to explore is "Complex Analysis" by Lars Ahlfors. It uncovers the beauty and applications of complex numbers in beautiful ways, perfectly blending mathematical theory with practical implications!