Question

Mario has a business selling muffins. Let x be the price of a muffin. Then, the profit P for Mario’s business is given by p(x)=-100x2+350x-150 Choose the inequality that shows the business will make a positive profit.

Answer 1

The first question's answer is:

0<-100x^2+350x-150

The second question "Look at the factorization shown below.

0 < –100x2 + 350x – 150

0 < –50(2x2 – 7x + 3)

0 < –50(2x – 1)(x – 3)

Select the range that Mario can choose from to price his muffins and make a positive profit.

The answer is:

$0.50 < x < $3.00

Answer 2

To make a positive profit p(x)>0 we need to make:
$-100 x^{2} +350x-150 \ \textgreater \ 0$

Now we solve this for x:
$-2 x^{2} +7x-3 \ \textgreater \ 0$

We have:
a = -2
b = 7
c = -3

We will use formula for quadratic equation:
$x_{1} = \frac{-b+ \sqrt{ b^{2}-4ac } }{2a} \\ \\ x_{2} = \frac{-b- \sqrt{ b^{2}-4ac } }{2a} \\ \\ x_{1} = \frac{-7+ \sqrt{ 49-24 } }{-4} = \frac{-7+5 }{-4} = \frac{-2 }{-4} = \frac{1}{2} \\ \\ x_{2} = \frac{-7- \sqrt{ 49-24 } }{-4} = \frac{-7-5 }{-4} = \frac{-12 }{-4} = 3$

We got two solutions. One is fraction other is whole number. We will not consider fraction because the amount of muffins sold must be whole number. So our solution is:
x>3