Question

For this adventure, you and world renowned Professor Sherlock McMerlock are traveling to the Lost Island of Laplaya. When the boat arrives on the island shore, you and Professor McMerlock disembark on your adventure. You trudge through the jungles and arrive at three impressively large doors. About eye level on each door is an intricately carved keyhole. Directions are scratched into the wood above each keyhole.
To open these doors, you must speak three functions in standard form.
(see below for more)

Answer 1

One function, f(x), with two real rational solutions.
f(x) = x^2 - 4
One function, g(x), with two real irrational solutions.
f(x) =
One function, h(x), with two complex solutions.

Create these three functions and explain to Professor McMerlock how you know these functions meet each condition. Remember, he is a Professor so use complete sentences. (Hint: Make sure that the b is even on g(x).)

Like magic, the doors creak open with a burst of dust, musty air, and shrieking bats. You lead McMerlock down the first corridor. It is so dark that you can barely see your own out–stretched hand. The darkness consumes you as you continue to trek deeper and deeper in the hallway. Suddenly, a mystical suit of armor blocks your path. A voice emanates from inside it.

“To pass by me you must tell me how to convert standard form into the general, vertex form... I have a test on it next week.”
Explain how to convert f(x) into the general, vertex form of the equation. Use complete sentences. You may use the f(x) you created in question 1 as an example.

Satisfied with his notes, the guardian lets you pass into the next chamber. As you enter the dimly lit room, McMerlock points out that the floor tiles have numbers on them, and that some floor tiles are missing. On the ceiling is painted a cryptic message.

“Only the solutions of g(x) will lead you safely across.”
Find the solutions of g(x). Show each step.
You can feel the treasure of the Lost Island of Laplaya in your grasp, as you deftly step on the correctly numbered stones. The last test of knowledge lay before you and Professor McMerlock. A wise old woman sits in front of the treasure. You can hear the crackle of magic coursing through her finger tips. She asks you “Is completing the square a good method for solving when the Discriminant is negative?” Professor McMerlock stammers, but you are confident in your answer.

Justify if completing the square is a good method for solving when the Discriminant is negative. Use any of your three functions as an example and respond in complete sentences.
Satisfied with your answer, the wise old woman grants you the treasure of the Lost Island of Laplaya! The treasure is an awesome ability to complete the square! Way to go! Professor McMerlock is thankful he picked you and promises to call on you again for another adventure.