Convert the equation to the standard form for a parabola by

Convert the equation to the standard form for a parabola by completing the square on x or y as appropriate x^2 - 6x - 6y - 21 = 0

given

x^2 -6x-6y-21=0

The standard form is (x - h)² = 4p (y - k), where the focus is (h, k + p) and the directrix is y = k - p

take 6y to the right hand side

then we get

x^2 -6x-21 =6y

x^2 -2.3.x -21 =6y ( 6 is written as 2*3 , because to make it in (x-3)^2 form )

now add and subtract (3)^2

x^2 -2.3.x +(3)^2 -(3)^2 -21=0

now [ x^2 -2.3.x +3^2 can be written as (x-)^2 ] so

(x-3)^2 -9-21=6y

(x-3)^2 -30=6y

(x-3)^2 = 6y +30

(x-3)^2 = 6(y+5)

the standard form of parabaola is (x-3)^2 =6(y+5)