1. By factorising – this is the simplest method provided that the quadratic can be factorised.                           Example: $2x^2-5x-3=0\Rightarrow(2x+1)(x-3)=0$$\Rightarrow x=-\frac{1}{2}, x=3$.
The quadratic formula says that if $ax^2+bx+c=0$ then the roots are given by $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$.
3. Completing the squarecompleting the square this is another infallible method for finding roots if a quadratic can be solved.   Example$x^2+6x+5=0\Rightarrow(x+3)^2-4=0$$\Rightarrow x+3=\pm 2\Rightarrow x=-1, x=-5$
Completing the square is also useful for sketching a quadratic. The reason for this is that, by writing the quadratic in completed square form, we can see the transformations applied to the graph of $x^2$. For example, $y=(x-3)^2+1$ is the graph of $x^2$ shifted to the left by 3 (x transformation) and then up by 1 (y transformation).