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Published 2018
This resource is part of our Adventures with Complex Numbers collection
When you first learned about quadratic equations, you may have talked about equations with no real roots. No real roots...
Take a look at the video showing the graph of $y = x^2 - 6x +c$:
When $c>9$ the red dots disappear.
But what if those two red dots are still around somewhere?
Where could they be?
Is there a mathematical answer to that question that makes sense?
Let's think a little more about $y=x^2-6x+c$.
If we complete the square, we can write it as $y=(x-3)^2 - 9+c$.
The roots are the solutions to $(x-3)^2-9+c=0$,
which we can solve by rearranging to give $(x-3)^2=9-c$.
That's fine if $c \leq 9$, but if $c>9$ we would have to find the square root of a negative number, and we can't do that! Or can we?...
For a little more background on how our number system expands in order to solve a wider variety of equations, click below and read on...
To explore Complex Numbers, the mathematical idea that explains where the roots go, try Opening the Door.