A piece of equipment has cost function C(x)=50x^2 + 1000 and its revenue
function is R(x)= 500x - x^2, where x is in thousands of items. What is the
least number of items that must be sold in order to break even?
50x^2 + 1000 = 500x - x^2
50x^2 + x^2 + 500= 0
51x^2 + 500 = 0
Do I just do -b/2a = -500/102= -250/51? What do I do next?
well, you should see right off that since 51x^2 is always positive, you can never have 51x^2 = -500.
So, let's see what happened.
50x^2 + 1000 = 500x - x^2
51x^2 - 500x + 1000 = 0
Now just use the quadratic formula to find the roots.
wow that was a silly mistake I didn't see the x next to 500 and thought I could just subtract it from 1000
thank you!
To find the least number of items that must be sold in order to break even, you need to solve the equation 51x^2 + 500 = 0.
To solve this quadratic equation, we can rewrite it in standard form by rearranging the terms: 51x^2 + 500 = 0.
Now we can see that a = 51, b = 0, and c = 500.
Using the quadratic formula, x = (-b ± √(b^2 - 4ac)) / (2a), we can substitute the values of a, b, and c into the formula.
x = (0 ± √((0)^2 - 4(51)(500))) / (2(51)).
Simplifying the equation further, we get:
x = ± √(-4(51)(500)) / (102).
Since we cannot take the square root of a negative number in real numbers, there are no real solutions to this equation.
Therefore, the least number of items that must be sold in order to break even is not possible.