Complex Solutions in Quadratics Shown Graphically

Do your Algebra 2 students struggle to understand imaginary numbers, complex numbers and imaginary zeros when graphing and solving quadratics? In this post is a link to a free download of a cheat sheet for showing students what complex zeros look like.
"So they're called imaginary numbers but they never should have been called that because they're super important for things like explaining the way electricity moves."

"Ummm, hmmm...."

Have you ever had a conversation with your Algebra 2 students that went something like this? I have! 

I recently came across a paper written by Carmen Melliger called How to Graphically Interpret the Complex Roots of a Quadratic Equation that brings alive what complex solutions to quadratics look like when graphed. The image on the second to last page is especially awesome because it magically shows us just what these complex solutions look like and how we arrive at them - all with no words!

I wanted to make a math cheat sheet for students so that they could see what complex solutions in quadratics look like. I used Ms. Melliger's paper as a guide to make a super rough 3-d model so that I could better visualize a drawing.

Then a rough drawing.

Complex Solutions in Quadratics Shown Graphically Cheat Sheet

And finally after a lot of trial and error, this image of the solutions to y = x^2 + 4x + 5 shown graphed.

Complex Solutions in Quadratics Shown Graphically Cheat Sheet

I added in a starting image of what the parabola looks like when graphed in the real plane and the solutions we get from the Quadratic Formula.

reflective Algebra 2 homework sheets

The first unit in this set of Algebra 2 homework sheets covers imaginary numbers and is based on the Common Core standards. 

You can find the complex solutions in quadratics cheat sheet here.

Update! An instagrammer sent me a link about Math teacher Philip Lloyd and his research on phantom graphs. So cool! 



  1. That is cool - the graph of the complex roots - I never knew that :)

    1. I think so, too. It’s cool to know they are graphable. Definitely a newer discovery for me, too:)