By making use of the imaginary number
*i* we can solve equations that involve the square
roots of negative numbers.

Complex numbers enable us to solve equations that we wouldn't be able to otherwise solve. Examine the following example:

$ x^2 = -11 \\ x = \sqrt{ \red - 11} \\ \sqrt{ 11 \cdot \red - 1} = \sqrt{11} \cdot i \\ i \sqrt{11} $

Without the ability to take the square root of a negative number we would not be able to solve these kinds of problems.