32. \( 2 x^{2}+7=-29 \)

To solve the equation \( 2x^{2} + 7 = -29 \), first, isolate the \( x^{2} \) term by moving 7 to the other side, which changes the equation to \( 2x^{2} = -29 - 7 \). This simplifies to \( 2x^{2} = -36 \). Next, divide both sides by 2 to get \( x^{2} = -18 \). Since the square of a real number cannot be negative, this indicates that the solutions will be complex. You can express the final answer as \( x = \pm \sqrt{-18} = \pm 3i\sqrt{2} \). To make it extra fun, you just crossed paths with imaginary numbers! They’re like the superheroes of the math world, swooping in to save equations from the villainous clutches of negativity. So, while most numbers can’t handle a negative under a square root, the imaginary unit \( i \) swings in, giving us fabulous roots like \( \pm 3i\sqrt{2} \)! On another note, don't be discouraged by complex solutions; they pop up often in higher mathematics, especially in fields like electrical engineering and quantum physics. Understanding complex numbers opens the door to a multitude of practical applications, from the analysis of electrical circuits to wave functions in quantum mechanics. Embrace the imaginary!