… Do you know why they exist then? …
Mathematicians are interested in finding all polynomial roots, so they want to solve for f(x)=0 even when a polynomial's graph doesn't touch or cross the x-axis. This is one reason why the imaginary unit was developed, to find all solutions (Real or not). You may google the subject, for more information.
What is the point of having the 'imaginary roots'?
That's a good research question for you! Google its variations, to find specific information like:
"Imaginary numbers are particularly applicable in electricity, specifically alternating current (AC) electronics. AC electricity changes between positive and negative in a sine wave. Combining AC currents can be very difficult because they may not match properly on the waves. Using imaginary currents and real numbers helps those working with AC electricity do the calculations and avoid electrocution."
"Imaginary numbers can also be applied to signal processing, which is useful in cellular technology and wireless technologies, as well as radar and even biology (brain waves). Essentially, if what is being measured relies on a sine or cosine wave, the imaginary number is used."
"Engineers employ imaginary roots because it makes the math easier."
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