Question

A company's monthly profit, P, from a product is given by P = -x2 + 105x - 1050, where x is the price of the product in dollars. What is the lowest price of the product, in dollars, that gives a monthly profit of $1,550? (Do not put a $ in your answer.)

Answer 1

This is one of those problems where you just have to look at what you have and what you are solving for. If you have a profit of 1550, then that means you have a value of P in the equation of 1550 like so: \[1550=-x^2+105x-1050\]

Answer 2

Now, you need to put the quadratic equation in a form that you can factor and solve, or you may use any number of methods to solve it (I like the quadratic formula), but lets see if we can factor it.

Answer 3

You always want it to be in the form 0=\[ax^2+bx+c\]

Answer 4

To do that, simply subtract 1550 from both sides.

Answer 5

\[-x^2+105x-2600\]=0

Answer 6

Multiply both sides by (-1) to make the leading term positive (just to make the math a bit easier in the end when we solve this thing), \[x^2-105x+2600=0\]

Answer 7

Give me a second to see if I can factor this.

Answer 8

I dont think it would be easy to do, I would solve with quadratic formula.

Answer 9

So, using the quadratic formula, we know that a = 1, b = -105 and c = 2600

Answer 10

The quadratic formula becomes: \[(-(-105)\pm \sqrt{(-105)^2-4(1)(2600)})/2(1)\]

Answer 11

= x, and the problem asks you to choose the smallest value of x

Answer 12

so the two solutions are x = \[(105+\sqrt{(-105)^2-(4)(1)(2600)} )/2\]

Answer 13

And \[(105-\sqrt{(-105)^2-(4)(1)(2600)} )/2\]

Answer 14

I gotta figure out how to type up fractions with this thing

Answer 15

\[\frac{(105+\sqrt{(-105)^2-(4)(1)(2600)}}{2}\]

Answer 16

And \[\frac{(105-\sqrt{(-105)^2-(4)(1)(2600)} )}{2}\]

Answer 17

Pick the smallest answer, and see if it works. Does this make sense?

Answer 18

If it does not work, I may have messed the arithmetic up

Answer 19

to me it dont sense

Answer 20

Are you familiar with the quadratic formula?

Answer 21

not really

Answer 22

Oh, well I can give you a quick intro on it if you would like.

Answer 23

okay

Answer 24

So what it is, is it is a way of solving equations that are in the following form:
\[ax^2+bx+c=0\]

Answer 25

An example of an equation in this form is something like:
\[2x^2+3x-9=0\]

Answer 26

So let me ask you a quick question to make sure you are up to speed. If I have an equation \[3x^2-5x+6=0\]

Answer 27

What are the values of a, b, and c?

Answer 28

since it is in the form \[ax^2+bx+c=0\]

Answer 29

a=3 b=5 c=6

Answer 30

Very close :), in this example you have to be careful, because the value of b is actually -5, not positive 5.

Answer 31

right, because it is really \[3x^2+(-5)x+6=0\]

Answer 32

ohh okay

Answer 33

The other two are correct.

Answer 34

Cool, so the quadratic equation is just a big fancy formula that was figured out. And if you know the values of a, b, and c, you can just plug them in and you will get the solutions of x.

Answer 35

It is in this form:

Answer 36

x=\[\frac{-b \pm \sqrt{b^2-4ac}}{2a}\]

Answer 37

Have you seen this before in your class?

Answer 38

yes

Answer 39

Ok cool, lets do an easy example of how you plug in the values to this thing. So here is another equation in the form \[ax^2+bx+c=0\]

Answer 40

You tell me what a,b, and c are.
\[2x^2+4x-1=0\]

Answer 41

a=2b=4c=-1

Answer 42

Yup, good. Now here is how you plug them into the formula.

Answer 43

\[x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}=\frac{-4\pm \sqrt{(4)^2-4(2)(-1)}}{2(2)}\]

Answer 44

Look at it very closely, where there was a "b", I just plugged in the value that we got for b in the equation, and the same for the values of a and c

Answer 45

Once you understand that, I will tell you how to deal with the

Answer 46

plus or minus sign

Answer 47

i get that

Answer 48

Ok, most people get tripped up by the plus or minus sign, and all it means is that there are two ways to write it. You can write the solution with the plus sign, and you can write the solution with the minus sign. So the one that we just found, to get rid of the plus or minus sign, just split it up like so:
\[x=\frac{-4+\sqrt{(4)^2-4(2)(1)}}{2(2)}\]

Answer 49

And \[x=\frac{-4-\sqrt{(4)^2-4(2)(1)}}{2(2)}\]

Answer 50

Thats all there is to it, I recommend always when you substitute the values of a, b, and c into the equation that you put them in parenthesis because remember that for example \[-2^2\neq(-2)^2\]

Answer 51

Now you would just have to simplify the answers to get x

Answer 52

okay now i get it thank you

Answer 53

\[x=\frac{-4+\sqrt{8}}{4}=-1+\frac{\sqrt{8}}{4}\]

Answer 54

and the same for the other solution

Answer 55

Awesome, I hope you do. That is all you need to do for your problem, the only trick is getting it in the right form.

Answer 56

Try your problem now, and let me know if you have any problems