Question

Find the coordinates of the circle and the radius y²+2x+x²-24y= -120

Answer 1

we know that
the equation of a circle is
(x-h+(y-k)²=r²
where (k,k)----> is the center
and
r----> is the radius
y²+2x+x²-24y= -120

Group terms that contain the same variable
(y²-24y)+(x²+2x)=-120

Complete the square twice. Remember to balance the equation by adding the same constants to each side
(y²-24y+144)+(x²+2x+1)=-120+144+1

Rewrite as perfect squares

(y-12)²+(x+1)²=25

(x+1)²+(y-12)²=5²

the answer is
the coordinates of the center are (-1,12)
the radius is 5 units