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Understanding how to solve quadratic equations is essential for students in algebra. Quadratic equations are expressed in the form of ax^2 + bx + c = 0, where a, b, and c are coefficients and x represents the unknown. To find the zeros of a quadratic function, like in the exercise g(x)=9x^2+81, one must first set the function equal to zero.

After setting the equation to zero, you can use various methods to solve for x. These include factoring, completing the square, using the quadratic formula, or, like in the provided solution, simplifying the equation step by step. Subtracting 81 and dividing by the coefficient 9 simplifies the equation to x^2 = -9, which serves as a foundation to deal with the next concept of imaginary numbers.

Imaginary numbers may seem daunting, but they are a fundamental part of understanding complex numbers. Imaginary numbers are necessary when trying to take the square root of a negative number, as the solution does not exist within the set of real numbers. The unit for imaginary numbers is i, where i = \(\sqrt{-1}\). In our exercise, when you encounter x^2 = -9, this simplifies to x = \sqrt{-9}.

You split \sqrt{-9} into \sqrt{9} * \sqrt{-1} which further simplifies to 3i, introducing the imaginary component. Imaginary numbers work like real numbers in many ways, allowing for addition, subtraction, multiplication, and division, expanding our ability to solve equations that have no real number solutions.

Square root properties are crucial when working with quadratic equations. The square root of a number x is a value that, when multiplied by itself, gives x. Importantly, every positive real number has two square roots: one positive and one negative. This is why we consider both +3i and -3i as solutions to x^2 = -9.

The concept of square roots also extends to imaginary numbers. When taking the square root of a negative number, as done in this exercise, we involve the imaginary unit i. Remember though that squaring an imaginary number will result in a negative real number. For example, (3i)^2 = 9i^2 = 9(-1) = -9, which brings us full circle to the original equation x^2 = -9.