For now I'm going to spend the rest of this post with a nice bit of proof showing how we get the quadratic equation from.

If you look online there are many long winded ways of showing proof for this formula. Most take up a page and can be quite hard to follow: I prefer this method.

Like with most quadratic formulas, we start with the basic idea that \[ax^{2}+bx+c=0\] From here, we can start by multiplying through by $4a$

So now we have an equation that looks a little something like this: \[\left(2ax+b \right)^{2}-b^{2}+4ac=0\]

Taking $-b^{2}+4ac$ over to the other side, then square rooting leaves us with:\[2ax+b=\pm\left(b^{2}-4ac\right)\] Now all that is left to do is minus the $b$ and divide through by the $2a$, leaving us with the oh so lovely: \[x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\]

Beautiful