**The Quadratic Equation**

Author: James Lowman

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:

\( ax^2 + bx + c = 0 \) (1)

\( x=\frac{-b \pm \sqrt{b^2 – 4ac}}{2a} \) (2)

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 hav 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. Take, for example, the following quadratic:

\( 10x^2 + 13x – 30 = 0 \) (3)

Some people may be able to see immediately that this equation can be factored. Others might have enough experience with factoring to only fumble through one or two permutations before generating the factored result quickly. The rest of us, myself included, might struggle with 5 or more permutations before coming close to a factored result. Yet the quadratic yields the answer in 4 quick lines:

\( x=\frac{-13 \pm \sqrt{169 + 1200}}{20} \) (4)

\( x=\frac{-13 \pm \sqrt{1369}}{20} \) (5)

\( x=\frac{-13 \pm 37}{20} \) (6)

so \( x= \frac{6}{5} \) and \( x=\frac{-5}{2} \) (7)

The quadratic equation is always the answer.