# Mathematics and Formulas

I remember the first time I was told about the quadratic formula. I was mind blown! How could someone possibly come up with this formula? If you haven’t seen it, the quadratic formula gives us the solutions to a (quadratic) equation $ax^2+bx+c=0$ (here $a,b, c$ are fixed numbers and $x$ is the value we want to solve for). The quadratic formula tells us that $x$ is given by

$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}.$

For example, if $x^2-5x+6=0$ (so $a=1, b=-5, c=6$), then there are two possible values that $x$ can be:

$x=\frac{-(-5)+\sqrt{(-5)^2-4\cdot 1\cdot 6}}{2\cdot 1}=3$, or $x=\frac{-(-5)-\sqrt{(-5)^2-4\cdot 1\cdot 6}}{2\cdot 1}=2$

It seems impossible that someone could just sit down and think up this formula, and it is! The problem with formulas in mathematics is that they (usually) don’t give you an understanding of how something works. Formulas should come after understanding. Mathematics is not supposed to be an exercise in memorizing lots of formulas; it is supposed to be a tool to help you understand and simplify complex problems. Formulas are then just a convenient language in which to state your results.

#### Why you shouldn’t just memorize formulas!

Unfortunately, the idea of understanding a problem before having a solution is usually under-emphasized. Because most of our mathematics education is exam-focused, it can be tempting to just memorize some formulas and learn which formulas to apply for each type of question that might come up. Whilst this can work in an exam situation, you will almost certainly not be able to solve unfamiliar problems.

As an example, whilst the quadratic formula above lets you solve any equation of the form $ax^2+bx+c=0$, adding an $x^3$ term will render it completely useless. The quadratic formula offers no insight into the problem of solving $ax^3+bx^2+cx+d=0$. However, understanding how to solve a quadratic equation will help you with this (much) harder equation: the first steps for each are essentially the same.

#### How do you memorize formulas?

Whilst you don’t need to memorize lots of formulas to be good at math, there will be some that you do need to remember. This should be a relatively small list, however (as an example, I can fit all the trigonometry you need to know for high school on half an A4 page, with large handwriting!). Once you know exactly which formulas you should be memorizing, the best way to commit them to memory is to do problems and write the formula down every time you use it (don’t just look it up or have it sitting in front of you). This has two benefits:

1. You will write the formula down so many times you won’t be able to forget it.
2. You will be practicing exactly the working out that you need to provide in an exam to get full communication marks.

I’ve found this to be one of the most efficient methods of studying for mathematics exams. If you need help with any of the above, feel free to ask!

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