<s>[INST] Derivation of the quadratic equation So everyone knows that when $ax^2+bx+c=0$,$$x = \dfrac{-b\pm\sqrt{b^2-4ac}}{2a}.$$
But why does it equal this? I learned this in maths not 2 weeks ago and it makes no sense to me
 [/INST] $ax^2+bx+c = a(x^2+\frac{bx}{a}+\frac{c}{a})=a((x+\frac{b}{2a})^2+\frac{c}{a}-\frac{b^2}{4a^2})$
Equate this to zero
NOTE: $a\neq0, $ if it is then the equation will not be quadratic.
you get $(x+\frac{b}{2a})^2=\frac{b^2}{4a^2}-\frac{4ac}{4a^2}$
which gives the desired $$x = \dfrac{-b\pm\sqrt{b^2-4ac}}{2a}.$$
</s>