Alg 2 - Quadratics - a favorite question

Again, in Accelerated Algebra 2, I made new graded assignment sheets of 10 questions each, on any type of material.  In Accelerated Algebra 1, the students are introduced to the quadratic formula and imaginary numbers.  In Accelerated Algebra 2, we haven't reviewed this yet, so it was interesting to see what they remembered from Algebra 1 with this question.

This is another one of my favorite questions from a different assignment sheet.  Again, I apologize, I don't remember where I found it:



So much goodness - what is a root?  What is a quadratic equation?  What does "integral coefficients mean"?  And, then is that number one number or two numbers?

When I started doing this one myself, I started by thinking if this is complex root, there is another root that is subtracting the imaginary part and I can write these as factors and multiply.  It got messy.  I think I made a mistake and I was not successful, so I tried another method.

I decided to work through the quadratic formula backwards.  I saw a common denominator of 40 and worked from there.  It worked pretty well.  I could figure out "a" and "b" but c was still a fraction.  Some kids who worked through this wanted to hand it in with integer A, integer B, and fraction C.  Nope, what can they do?  They figured it out.  (I am not going to provide answers here.)  I was successful in this method though and really liked it - giving thought to how do you work with the "i" backwards, how do you work with number outside the square root backwards?

I also tried factoring backwards.  If I use "a" as one and try to factor forwards with the "what multiplies to "c" and adds to "b"?  Then, I could do that backwards.  I added them to get "b" and multiplied them to get "c".  I dealt with the fractions and voila!

Three great methods for this problem.  As I worked with students, I asked them some guiding questions to see what they remembered, decide what "root" they wanted to take (pun intended).

I like it!