What is the equation of a sphere in standard form?

The answer is: ##x^2+y^2+z^2+ax+by+cz+d=0##,

This is because the sphere is the locus of all points ##P(x,y,z)## in the space whose distance from ##C(x_c,y_c,z_c)## is equal to r.

So we can use the formula of distance from ##P## to ##C##, that says:

##sqrt((x-x_c)^2+(y-y_c)^2+(z-z_c)^2)=r## and so:

##(x-x_c)^2+(y-y_c)^2+(z-z_c)^2=r^2##,

##x^2+2(x)(x_c) + x_c^2+y^2+2(y)(y_c)+y_c^2+z^2+2(z)(z_c)+z_c^2=r^2##,

##x^2+y^2+z^2+ax+by+cz+d=0##,

in which

##a=2x_c##; ##b=2y_c##; ##c=2z_c##; ##d=x_c^2+y_c^2+z_c^2-r^2##;

So:

##C(-a/2,-b/2,-c/2)##

and ##r##, if it exists, is:

##r=sqrt(x_c^2+y_c^2+z_c^2-d)##.

If the center is in the Origin, than the equation is:

##x^2+y^2+z^2=r^2##,