Susan is selling tickets to a concert. Seating capacity is 1,500 seats. The
rate is $6.00. Susan knows that she can sell 1200 at this price. Every time
that she increases the price by $0.75 then her sales decreases by half. write
the equation. What is the maximum amount that she can make.

Your teacher botched this problem.  First of all it is a calculus "minimax"
problem, not an algebra problem, since it is not a quadratic.  Secondly Susan
loses money if she increases the price even by $0.75, for only half the 1500
seats, or 750, are then sold, and that only brings in $5062.50, whereas if
she doesn't increase it at all, she takes in $9000.

If she increases    The price per      And the number      So her
 the price by       ticket is      of ticket sales is   intake is

      0              $6.00               1500            $9000.00
    $0.75            $6.75                750            $5062.50
    $1.50            $7.50                325            $2437.50
    $2.25            $8.25                162            $1336.50
    $3.00            $9.00                 81            $ 729.00
    $3.75            $9.75                 40            $ 390.00
    $4.50           $10.50                 20            $ 210.00
    $5.25           $11.25                 10            $ 112.00
    $6.00           $12.00                  5            $  60.00
    $6.75           $12.75                  2            $  25.50
    $7.50           $13.50                  1            $  13.50
    $8.25           $14.25                  0            $   0.00

So she makes the most money by not increasing the price at all.
     
Tell your teacher he or she has botched this problem.

Here is an attempt at deriving the equation for it:

Let x = the number of $0.75 price increases

Then the price per ticket is $6.00 + $0.75x

>>..Every time that she increases the price by $0.75 then her sales 
decreases by half..<<

So x is also the number of times her sales get cut in half, so
the 1500 is multiplied by  x times.

So the number of sales is 1500

So her total revenue R = (price per ticket)(number of tickets sold)

R = (6 + .75x)(1500

So you see it is an exponential equation, not a quadratic, which
is probably what your teacher intended it to be.

Edwin