Explanation:

Given:
- Focal length of the concave mirror (f) = -15 cm (negative sign indicates concave mirror)
- Image formed is real, which means it is formed on the same side as the object
- There are three types of images formed: real, inverted, same size

To calculate the distance of the object from the mirror, we can use the mirror formula:

1/f = 1/v - 1/u

Where:
- f is the focal length of the mirror
- v is the image distance from the mirror (negative for real image)
- u is the object distance from the mirror (negative for real object)

Step 1: Determine the image distance (v)
Since the image formed is real, the image distance is negative. Let's assume v = -x cm.

Step 2: Determine the object distance (u)
Since the object is kept in front of the mirror, the object distance is positive. Let's assume u = y cm.

Step 3: Apply the mirror formula and solve for u
1/f = 1/v - 1/u

Substituting the given values:
1/-15 = 1/(-x) - 1/y

Simplifying the equation:
-1/15 = -y/xy - x/xy

Multiplying both sides by 15xy:
-xy = 15y - 15x

Rearranging the equation:
15y - xy - 15x = 0

Step 4: Find the size of the object (height)
The size of the object is the same as the size of the image in this case. Let's assume the height of the object is h cm and the height of the image is also h cm.

Step 5: Apply the magnification formula and solve for h
Magnification (m) = -v/u = h'/h
Since the image is inverted and the height is the same, the magnification is negative.

Substituting the values:
-m = -x/y = h/h

Simplifying the equation:
x = y

Step 6: Solve the equations simultaneously
We have two equations:
15y - xy - 15x = 0 (from step 3)
x = y (from step 5)

Substituting x = y in the first equation:
15y - y^2 - 15y = 0

Simplifying the equation:
y^2 - 30y = 0

Factoring the equation:
y(y - 30) = 0

So, y = 0 or y = 30

Since the object cannot be at a distance of 0 cm from the mirror, the object distance is y = 30 cm.

Therefore, the distance of the object from the mirror is 30 cm.