Kool Klothes has determined that the revenue function for selling x thousand pairs of shorts is R(x)=-5x^2 + 21x. The cost function C(x) =2x +10 is the cost of producing the shorts.

Write a profit Fumction
How many pairs of shorts must the company sell in order to break even

p = R - C

= 5 x^2 + 19 x - 10

where is that zero?
solve quadratic
x = [ -19 +/- sqrt (361+200) ]/10

= [ -19 +/- 23.7 ]/10

= .469 thousand = 469

To write the profit function, we need to subtract the cost function from the revenue function. The profit function, P(x), can be found by subtracting C(x) from R(x):

P(x) = R(x) - C(x)

Given that R(x) = -5x^2 + 21x and C(x) = 2x + 10, we can substitute these values into the equation to obtain the profit function:

P(x) = (-5x^2 + 21x) - (2x + 10)
= -5x^2 + 21x - 2x - 10
= -5x^2 + 19x - 10

To find the number of pairs of shorts that the company must sell in order to break even, we need to set the profit function equal to zero:

P(x) = 0

-5x^2 + 19x - 10 = 0

To solve this quadratic equation, we can either factor, complete the square, or use the quadratic formula. Let's use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

In this case, a = -5, b = 19, and c = -10. Substituting these values into the formula:

x = (-(19) ± √((19)^2 - 4(-5)(-10))) / (2(-5))
= (-19 ± √(361 - 200)) / (-10)
= (-19 ± √161) / (-10)

Therefore, the company must sell approximately (-19 + √161) / (-10) or (-19 - √161) / (-10) thousand pairs of shorts in order to break even.