Deriving the Quadratic Formula – A Subtlety

I ask my intermediate algebra students to derive the quadratic formula on their last test and on the final.  This requirement has the same utility as asking a literature student to memorize a poem.  It can’t be done without some measure of understanding.  I tell the class  that learning to derive the quadratic formula will change their lives – their algebra lives at least.  The derivation isn’t worth many points on the exams but I think all ‘A’ students should be able to do it.  By the way I check their calculator’s memory to be sure they are not just transcribing.

I expect the work to look something like this without the comments on the right.

Derivation of the Quadratic Formula

The steps are correct but look carefully at the two steps in the red box.  They mask a  subtle argument.  Actually \sqrt{4a^2} = 2|a| since we don’t know whether a is positive or negative.  Some reasoning along this line needs to be made.

Derivation of the QuadraticFormula – Subtlety

This can be  a little too much for the typical intermediate algebra student to take. The texts that I surveyed either ignore it altogether or work out the case where a > 0 and leave a < 0 for the exercises.  If I were to explain it I would first do a numerical example like solving 3x^2+7x - 11 = 0 say by completing the square and then I would solve -3x^2-7x+11=0 in the same way.  In the second case after we divide by -3, the steps are the same as the first case but when we get to the square root  in the actual derivation and compare it to the numerical examples we can see the error of our ways.  Below is a partial demonstration.

Quadratic Formula with Negative Second Degree Coefficient

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About jrh794

I am a sixty-five year old math instructor at Southern Oregon University. I taught at the College of the Siskiyous in Weed California for twenty-six years. Prior to that I worked as a computer programmer, carpenter and in various other jobs. I graduated from Rice University in 1967 and have a MS in Operations Research from Stanford. In the past I have hand-built a stone house and taken long solo bicycle tours. Now I ride my mountain bike and play golf for recreation.
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