The equation x² + 1 = 0 is a quadratic equation, which is a type of polynomial equation of degree two. To solve it, we can rearrange it to x² = -1. This indicates that x must be a number whose square is negative, which is not possible within the set of real numbers. To find solutions, we introduce the concept of imaginary numbers. The solutions to this equation are x = i and x = -i, where i is defined as the imaginary unit, equal to the square root of -1. This concept is fundamental in complex number theory, which extends the number system beyond real numbers.