The quadratic equation is always the answer. Time and time again, while tutoring students, I encounter a resistance to using an unfailing tool called the quadratic equation. This simple algebraic mathematical statement allows a student to find the roots of any clumsy second order polynomial with ease. I can only assume that, while the quadratic equation is drilled into memory, its useful nature is under-reported in high school mathematics.
For those who aren’t familiar:
If you can get an equation into the form of equation ?? , then you can use the coefficients a, b, and c in order to solve for x in equation ??. The ± in equation ?? implies that there are two solutions for x (this being the result of solving a second order polynomial equation). These two solutions represent the roots of equation?? The roots are the values of x that allow equation?? to equal 0 on the left hand side. Roots are of interest to us, because they are a quick and easy way to determine factors of a quadratic (not to mention all the mathematical reasons to identify when a function is equal to zero). Students are drilled on factoring. I rarely see a student that is unable to factor a simple quadratic, but I often encounter those that have problems when the factors fail to be immediately obvious. The quadratic equation is always the answer.
I often despair that education relies entirely on problem singularity, a belief that there is only one correct way to approach a question. When I see a student struggle for minutes to try and factor a difficult quadratic, I can’t help but wonder why they shy away from the quadratic equation. Sometimes it is hyper- focus that keeps them from finding an alternate path forward, but more and more I encounter fear. Factoring is supposed to be an easy shortcut, while the quadratic equation has the confusing ± and the scary √ . But a shortcut fails to be short when it takes time to muddle through possible variations.