vertex form is y = a(x - h)^{2} + k where (h,k) is the vertex

root form is (ax-c)(bx-d) where a and b do not equal a and b in standard form but are just coefficients

then we have the quadratic equation which is -b+ or - sqrt(b^2-4ac) all divided by 2a, this will give the roots

but to tell the type of root we use b^2-4ac and if its positive thre are 2 real roots, if its 0 there is one real root and if its negative they are complexed roots (meaning the quadratic does not touch the x-axis)