Completing the Square for Quadratics

$$\text{Sherlock Holmes Day}$$

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Today (May 22) is Sherlock Holmes Day!

A Study in Squares

Sherlock Holmes and Dr. Watson are investigating a crime scene, at a second floor bakery. Mr. Victum, the baker, was shot sometime the night before. The police cannot find the bullet in order to identify the gun used.

Sherlock notices a hole in the window and some glass on the floor inside. He also notices that the entry and exit wound on Mr. Victum indicates he was shot from below. Holmes deduces that the killer was outside the bakery and fired up at the late Mr. Victum.

Furthermore, there were footprints in the dust by he window. These matched the victim's shoes. Judging by the hole in the window, the wounds, and the height of Mr. Victum, Holmes figured the bullet followed a parabolic path of: $$h(t)=-\frac29t^2+\frac{16}9t+\frac{130}{9}$$ where $t$ is measured in feet.

Holmes figures that the bullet must have lodged in one of the many overhanging objects in the room. By completing the square, he is able to figure out the vertex of the parabola and correctly identify which of the objects the bullet is lodged in.

The following are the possible objects and their coordinates:

1. A Chandelier $(2,16)$
2. A Portrait of a Cinnamon Roll $(7,12)$
3. A Globe $(5,13)$
4. A Animal Head $(4,18)$