Personal Blogs

I'm trying to teach myself a bit of LaTeX, which if you don't know is a specialised language that allows a user to enter expressions as plain text and have them rendered as mathematical notation.

For a while I've felt not knowing LaTeX is a gap in my toolkit but the immediate prompt is that I'm writing accessibility guides for a couple of modules and I'm looking how someone who is blind or has low vision can read and write mathematical notation, and LaTeX is one route, especially for anyone planning to go on to study more mathematics. So as an exercise I decided to write a post about a neat mathematical trick I learned recently. 

My Exercise

Suppose we're asked to solve the following quadratic equation by factorisation

This is not too hard. We just look for a pair of numbers that multiply to give and add to . These are and add to . So the equation factorises to

and the solutions are and .

But what if the coefficient of is greater than ? For example we might have

Now we have to consider all the factors of and of and figure out which combination is the correct one and it all gets a bit messy.

Here's a neat trick that makes this equation as easy to solve as the first example. We say (first coefficient times last), set the first coefficient to and form a new equation (but remember that , we will need it later!)

This is easy to factorise and we get , so the solutions are and . Of course these are not solutions of the original equation but now we bring that back into play, and divide the two solutions we have just found by it (makes a kind of sense you see, first we multiplied by , now we divide by it).

and

and, surprise, surprise, these are the solutions to the original equation!

Why does this work?

Multiplying the original equation by will give

or .

Now replace by and we get the second equation and the roots of this must be times those of the original equation.