Rewritting equation in standard form of a quadratic equation, and solving quadratic, using formula?

Question

A retailer spent $48 to purchase a number of special mugs. Two of them were broken in the store, but by selling each of the reaming mugs for $3 above the original cost per mug, she made a total profit of $22. Construct an equation that will allow us to solve for the number of mugs, denoted by n, that were originally purchased
a. if the price for n mugs is $48, how can we express the cost per mug
b. We are told that the retailer, in order to make a specific profit, sells the remaining mugs for $3 more the the original cost per mug. Using this information, and your answer to step (a) how can we express the selling price per mug?
c. since two of the mugs were broken in the store, how can we represent the number of mugs that are available for sale?
d. Use the answers from step (b) and (c) to construct an expression for the amount of money made on the sale of the remaining mugs
e. Write an equation that states this relationship between the money made, the cost, and the profit

1. Rewrite equation you wrote in step (e) in standard form of a quadratic equation:
2. solve equation
3. How many mugs were purchased?
4. what was the origianal cost per mug?
5. what profit was made on each mug?
6. When solving quadratic equations there are normally 2 solutions, why are you able to discard one of the solutions in this case?

I know the first part works out something like this, but what about the rest im having trouble with the quadratic part
A.
48/n
b.
(48/n) + 3
c.
n - 2
d.
(n-2)*(3 + (48/n))
e.
22 = (n-2)(3 + (48/n)) - 48
70 = (n-2)(3 + (48/n))
70 = 3n - 6 + 48 - 96/n
28 = 3n - 96/n
28n = 3n^2 - 96
3n^2 - 28n - 96 = 0
(3n + 8)(n - 12) = 0
n = 12

Answer 1

Okay, I'll try to do the whole thing. Let n be the number of mugs purchased. She spent $48, so the cost per mug was 48/n (answer a).

She sold n-2 mugs (answer c), and her selling price was 3 + 48/n (answer b). So her revenue was (n-2)(3 + 48/n). Profit ($22) is revenue minus cost ($48), which can be written as

(n-2)(3 + 48/n) - 48 (answer d); and

(n-2)(3 + 48/n) - 48 = 22 (answer e)

Now you want to rewrite (e) in standard form:

3n + 48 - 6 - 96/n - 48 = 22 (multiplying the binomials)
3n - 28 - 96/n = 0 (collecting terms)
3n^2 - 28n - 96 = 0 (multiply by n to get in standard form -- answer 1)

(3n + 8)(n - 12) = 0 (factoring; note the middle term is -36+8 = -28)

n = -8/3; n = 12 (answer 2 -- solve the quadratic)

There were 12 mugs purchased. (Answer 3; we solved to find n = 12.)

The profit per mug was $22/12 = $1.83 (answer 5). Notice that we figured the unit profit based on the number of mugs purchased (12) rather than the number sold (10). Had we done it the other way, the answer would be $22/10 = $2.20.

And for answer 6, we discarded the negative value for n (-8/3). We knew that a positive whole number of mugs were purchased.

I think that's everything.

Oh -- one more thing. If you need to have that quadratic solved by the quadratic formula (even though it can be factored), that solution is

3n^2 - 28n - 96 = 0
n = {-(-28) +/- sqrt[28^2 - 4(3)(-96)]} / (2*3)
n = [28 +/- sqrt(28^2 + 12*96)] / 6
n = [28 +/- 4 sqrt(7^2 + 3*24)] / 6
n = [28 +/- 4 sqrt(121)] / 6
n = (28 +/- 4*11) / 6
n = (28 + 44) / 6 = 12 n = (28 - 44) / 6 = -8/3

I think you wanted the quadratic solution this way too. Notice that factoring (or completing the square) is easier.

Answer 2

let the no of mugs bought be n
cost/mug=48/n
no of mugs sold=n-2
selling price= no of mugs sold(n-2) multiplied by the S.P. of one mug{(48/n)+3}
but this is also equal to the S.P. which is equal to cost +profit (48+22) and so equating
(n-2)(48/n +3)=48+22=70
(n-2)(48+3n)=70n
48n-6n-96+3n^2=70n
rewriting in the std Q.E. form
3n^2-28n-96=0
3n^2-36n+8n-96=0
3n(n-12)+8(n-12)=0
if the product is zero either factor is zero
so n=12 or-8/3 which is not acceptable

cost price of the mug=48/12=$4
selling price=70/10=$7
profit /mug=$3
no of mugs bought=n and two got broken so saleable mugs is 2 less than the no bought=n-2
selling price per mug=revenue/no of mugs sold=70/(n-2)
revenue=C.P.+Profit=48+22=$70

Answer 3

a. cost = 48/n.

b. price = 3 + 48/n.

c. num = n - 2.

d. amount = (n - 2)*(3 + 48/n)

e. (n - 2)*(3 + 48/n) = 48 + 22, or (n - 2)*(3 + 48/n) = 70

after some math:

3*n^2 - 28*n - 96 = 0

and the positive root is

n = 12

Answer 4

the "quadradic" is nothing but the fact that your equation has an exponent of highest power of two....
when you reached your "factorization" of your solution..(3n^2-28n-96)... you could have used the quadradic equation to solve for your answer ( a=3.. b= -28 .. c= -96)

Answer 5

b. (n-2)(x+3) = 70

x+3 = 70/(n-2)

x = (70/(n-2)) - 3

Answer 6

I have no idea what you are referring to.