A number whose square is a negative number. That is the realm we are about to explore.
First, let's try to find an answer with the numbers we already know.
Let \(x\) be . Then we compute . If you run through any option you like, you will find that none of them give you a negative number.
So we make up a new number, a number called \(i\), which is also written as \(\sqrt{-1}\). This is a number whose defining property is that \(i^2 = -1\).
We define addition with \(i\) in the same way as if this was 1. For example, \(i+i = 2i\). This leads to statements like \(3*2i = 6i\) and \(3i*2i=-6\)
There ae two such numbers, namely \(i\) and \(-i\). We have chosen one of them aribitrarily to be \(i\). While it is largely not an issue to worry about, it can be useful to keep it in mind to avoid relying on an arbitrary choice (whatever you say about \(i\) should also apply to \(-i\)).
This solves the issue of negative square roots
When there is no solution, reflecting across the horizontal through the vertex leads to a parabola with solutions. Find those roots. Draw the circle through those roots with that being a diameter. Then the complex points in the plane for the original's roots are the points at the top
contrast with Carlyle circle for real roots
complex root circle construction explained: https://scholarworks.umt.edu/cgi/viewcontent.cgi?article=1440&context=tme