SOLUTION: The profit on a watch is given by P = x^2 - 13x -80, where x is the number of watches sold per day. How many watches were sold on a day when there was a $50 loss? Thanks so mu

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Question 76658: The profit on a watch is given by P = x^2 - 13x -80, where x is the number of watches sold per day. How many watches were sold on a day when there was a $50 loss?

Thanks so much for you help this one has me stummped.
Found 2 solutions by stargrl12566, bucky:
Answer by stargrl12566(16) About Me  (Show Source):
You can put this solution on YOUR website!
First you make your equation which is -50=x^2-13x-80
Then add 50 to both sides x^2-13x-30+0
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-13x%2B-30+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-13%29%5E2-4%2A1%2A-30=289.

Discriminant d=289 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--13%2B-sqrt%28+289+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-13%29%2Bsqrt%28+289+%29%29%2F2%5C1+=+15
x%5B2%5D+=+%28-%28-13%29-sqrt%28+289+%29%29%2F2%5C1+=+-2

Quadratic expression 1x%5E2%2B-13x%2B-30 can be factored:
1x%5E2%2B-13x%2B-30+=+1%28x-15%29%2A%28x--2%29
Again, the answer is: 15, -2. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-13%2Ax%2B-30+%29

Answer by bucky(2189) About Me  (Show Source):
You can put this solution on YOUR website!
Given:
.
P+=+x%5E2+-+13x+-80
.
Where P = profit and x = the number of watches sold in a day.
.
If there was a $50 loss for the day, the profit was -50. Substitute this amount of profit
into the equation and you get:
.
-50+=+x%5E2+-13x+-+80
.
Eliminate the -50 on the left side by adding +50 to both sides to get:
.
0+=+x%5E2+-+13x+-+30
.
and reverse sides so that you have the equation in the standard quadratic form:
.
x%5E2+-+13x+-+30+=+0
.
The left side of this equation can be factored to get:
.
%28x+-+15%29%2A%28x+%2B+2%29+=+0
.
and this equation will be true if either of the factors are zero because multiplying
anything by zero results in zero.
.
So solve for values of x that will make the left side zero by setting each factor, one
at a time equal to zero.
.
x+-+15+=+0 can be solved by adding 15 to both sides to get x+=+15
.
and
.
x+%2B+2+=+0 can be solved by subtracting 2 from both sides to get x=-2.
.
We can reject the second solution because what does it mean to sell -2 watches ... buy 2 watches?
.
So the answer is that if you only sell 15 watches in a day, you lose $50.
.
And just for fun, let's get an idea of how many watches you need to sell daily to break even.
You break even if your profit is zero, so we can write the equation:
.
x%5E2+-+13x+-+80+=+0
.
If you solve this using the quadratic formula (if you don't know this method yet, you
probably will soon) you will find that the left side of this equation will be zero if
x is equal to approximately 17.56 or if x = -4.56. Again we can ignore the negative
solution. So this tells us that if 17 watches are sold in a day, you lose a little money
and if 18 watches are sold, you make a little money.
.
Hope this analysis of the watch business gives you some insight into solving quadratic
equations. A tricky part was to recognize that a negative profit is a loss, so you needed
to substitute minus 50 dollars for the profit and then solve the equation for x.