Quadratic Expression Calculator 2
Bugs


calculateQuadraticFormula() calculates wrong results for double roots.  Watch your operator associativity!

if( temp == 0 ) {
    results.add(-b / 2 * a);}

The "exit" feature doesn't work.  The prompt would never be "exit", would it?

private static double promptUserUntilValidInput(String prompt){
    if (prompt.equals("exit")) {
        sc.close();
        System.exit(0);
    }



Design issues


Class names should be nouns, and the name should reflect their purpose.  I recommend QuadraticSolver as the class name.
The calculateQuadraticFormula() function could then be simply named solve().
The class should not hold a Scanner as a member variable, as that would violate the Single Responsibility Principle.  It can be a local variable in main() that gets passed to the prompting function.  (The scanner should be created using a try-with-resources block.)
Returning null from the solver is annoying for the caller to handle.  I suggest returning an empty set if there are no real roots.
I suggest returning two copies of the result if it is a double root, but that is a matter for debate.


Style issues


You assign the expression (b * b) - (4 * a * c) to a variable named temp.  Why not use a descriptive name discriminant?
The promptUserUntilValidInput() function could be better if you removed its tryParse() helper, I think.  (Furthermore, promptUser… is redundant — who else are you going to prompt?)

Also, since there is a System.exit() hidden inside, it needs to be documented.  (Actually, @rolfl's suggestion to throw an exception is better.)

/**
 * Prompts the user to enter a number, and retries until the input is a
 * valid double.  Calls System.exit(0) if the input is "exit", or if EOF
 * is encountered.
 */
private static double promptUntilValidInput(String prompt, Scanner sc) {
    do {
        try {
            System.out.print(prompt);
            String input = sc.nextLine();
            if ("exit".equals(input)) {
                System.exit(0);
            }
            return Double.parseDouble(input);
        } catch (NoSuchElementException eof) {
            System.out.println();
            System.exit(0);
        } catch (NumberFormatException retryOnBadInput) {
        }
    } while (true);
}