The demand function for a new product is P(x) = -5x + 22 where x is the number of items sold in thousands and p is the price in dollars.

Umm. Shouldn't that be called the price function?

a) state the corresponding revenue function (I'm pretty sure it's -5x^2 + 22x)

Yes, it is.

Because revenue is (demand)*(price).

b) Find the corresponding profit function(I'm having problem with that)

Profit = Revenue -Cost

So,

Profit(x) = (-5x^2 +22x) -(3x +15)

Profit(x) = -5x^2 +19x -15 ---------------------answer.

c) Complete the square to find the value that will maximize the profits.

-5x^2 +19x -15

= -5[x^2 +(19/5)x +3]

= -5[x^2 +(19/5)x +(19/10)^2 -(19/10)^2 +3]

= -5[(x +(19/10))^2 -(19/10)^2 +3]

So, the x at maximum profit is -19/10

What?

Negative demand? No product?

There is something wrong with your Question as posted.

Edit:

Oopps, my mistake!

It should have been:

-5x^2 +19x -15

= -5[x^2 -(19/5)x +3]

= -5[x^2 -(19/5)x +(19/10)^2 -(19/10)^2 +3]

= -5[(x -(19/10))^2 -(19/10)^2 +3]

So the x at maximum profit is 19/10 thousand = 1900 products

Therefore, to maximize profit, 1900 of the products must me made. ---------------answer.

d) Find the break even quantities.

At break-even, the profit is zero because the revenue equals the cost only.

So,

-5x^2 +22x = 3x +15

-5x^2 +22x -3x -15 = 0

-5x^2 +19x -15 = 0

Divide both sides by -5,

x^2 -3.8x +3 = 0

Use the Quadratic formula,

x = {3.8 +,-sqrt[(3.8)^2 -4(1)(3)]} / 2(1)

x = {3.8 +,-1.562} /2

x = 1.119 or 2.681 thousands product.

Therefore, it is break-even if 1,119 products or 2,681 products are made. -------------answer.