I have been doing stuff with quadratics over the last few weeks, although I find some of it hard, I have understood most of it better than all the other content I have learned over the last few months.

So, I have now been learning how to solve a quadratic equation of the form $ax^{2}+bx+c$ were a, b, and c are constants, if you can not factorize the equation then you can use a formula $x=\dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a}$

To solve the quadratic equation $x^{2}+6x+1=0$ find the constants so, a=1 (because it is $x^{2}$ b=6 and c=1.

If I sub these in to the following formula. $x=\dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a}$ $=\dfrac{-6\pm\sqrt{6^{2}-4\times1\times1}}{2\times1}$ $=\dfrac{-6\pm\sqrt{36-4}}{2}$ $=\dfrac{-6\pm\sqrt{32}}{2}$ $32=16\times2$ $=\dfrac{-6\pm4\sqrt{2}}{2}$ $\sqrt{32}=4\sqrt{2}$ $=\dfrac{-6}{2} \pm \dfrac{4\sqrt{2}}{2}=-3\pm2\sqrt{2}$

so the solutions to this equation are... $x=-3+ 2\sqrt{2}$ and $x=-3-2\sqrt{2}$



I do have another quadratic equation to solve in the same sort of way, I will add that in the next couple of days. Making sure all the typesetting for the equations is correct takes a bit of time of which I have very little nowadays.