This is part of the “Memorize or Melt” series, for more information go here if you haven’t read the introductory post on this topic.

Here is the quadratic formula that students are often told to recite and commit to memory. $$\frac{-b \pm \sqrt{b^{2}-4ac}}{2a}$$

• What is this? What is its purpose?
• Why are students made to memorize this?
• What will students learn by having committed this to memory?
• Why does the author have this memorized? How did he benefit from this?

The quadratic formula, as it is taught to students, is a neat formula that gives the solutions for \(x\) to $$ax^{2} + bx + c = 0$$ for \(a \neq 0\) and \(a,b,c \in \mathbb{R}\).

Students are made to memorize this because … I don’t know why. But they are. One can suppose that the goal / hope is that since solving quadratics (polynomials of degree 2) will come up often in a math student’s study, it is more efficient to have memorized how to solve than to have to re-derive the formula each time. That’s an understandable but flawed logic.

At around the time the student is taught the quadratic formula, she is also introduced to “factoring”, “completing the square”, “FOIL” (though this acronym is (thankfully) falling out of favor, but hopefully won’t be replaced by something even more gaudy), etc. Now, learning the how and why of factoring, completing the square, and expanding products of parenthetical statements is certainly useful as these will often come up in math problems of all levels of difficulty. The quadratic formula? Useless. Yes, useless. There is no reason, whatsoever, that anyone should have to memorize this formula. Every student of algebra should be able to easily solve a quadratic equation without having memorized the general form of its solution. In fact, I will go so far as to say, that having students memorize the quadratic formula only hurts them because they eventually forget how it came about.

Instead, it is far better to have students solve quadratics by factoring or completing the square than to have them memorize a solution to one class of equations. Here are some advantages:

• Converting \(x^{2} + bx + c = 0\) to \((x + p)^{2} + q = 0\) (where \(p\) and \(q\) are constants dependent on \(b,c\)) helps students visualize the solution (think parabolas).
• With respect to the above point, it will help students better understand parabolas in general when they have to work with them in any number of math, engineering, and science courses.
• It will give students the opportunity to strengthen some mechanics.
• Instructors can discuss the idea of an imaginary solution.
• Instructors can slowly introduce the notion of solving systems of equations or at least plant the seed for something else to learn. Keep the students striving for more!
• It will be far easier for instructors to transition into more general notions of root-solving. Instructors will also be able to slowly explain the notion of “roots of multiplicity” and where this comes up in complex analysis.
• It may increase a student’s curiosity. She may wonder what techniques exist for solving polynomials of degree three. Right now, the vast majority of math students believe that the quadratic formula is the be-all, end-all of root finding.

Every math instructor should simply junk the quadratic formula as a thing to show and force students to memorize. Instead, one better alternative is to reason out with (not at) students the notion of completing the square. Here’s how without making students memorize the procedure for completing the square (“divide \(b\) by two … yak yak yak”).

Download Solving quadratics