Secure the Drywall With Screws
Take the sheet over to the wall and place a screw at the first stud. Carefully place the new drywall over the thinner drywall to avoid breaking the paper on the existing drywall.
With the drywall T-square and a pencil, draw a line down the drywall where the stud falls (Image 1). Place a screw every 12-16" along the stud to secure the sheet to the wall (Image 2).
The drywall bit for the 1/4" drywall allows you to sink the screws just below the surface without breaking the paper.