A company has determined that the profit, in dollars, it can expect from the manufacture and sale of tennis racquets is given by P = −0.01x2 + 130x − 180,000. How many racquets should the company manufacture and sell to earn a profit of $174,900

well, just solve for x in

-0.01x^2 + 130x - 180000 = 174900
x = 3900 or 9100

To find out how many racquets the company should manufacture and sell to earn a profit of $174,900, we need to set the profit equation equal to the desired profit and solve for the value of x.

The profit equation is given by P = -0.01x^2 + 130x - 180,000, where P represents the profit in dollars, and x represents the number of racquets sold.

We can set up the equation as follows:
174,900 = -0.01x^2 + 130x - 180,000

Now, let's rearrange the equation to bring all terms to one side:
0.01x^2 - 130x + 180,000 - 174,900 = 0

Simplifying this equation gives us:
0.01x^2 - 130x + 5,100 = 0

To solve this quadratic equation, there are multiple methods, such as factoring, completing the square, or using the quadratic formula. In this case, let's use the quadratic formula, which states that for a quadratic equation of the form ax^2 + bx + c = 0, the solutions for x are given by:

x = (-b ± √(b^2 - 4ac)) / (2a)

Applying this formula to our equation, where a = 0.01, b = -130, and c = 5,100, we can calculate the values of x:

x = (-(-130) ± √((-130)^2 - 4(0.01)(5,100))) / (2(0.01))
x = (130 ± √(16,900 - 204)) / 0.02
x = (130 ± √16,696) / 0.02

Now, let's calculate the values inside the square root:
√16,696 ≈ 129.34

Substituting this value back into the equation, we have:
x = (130 ± 129.34) / 0.02

Now, we can calculate the two possible values for x:
1. x = (130 + 129.34) / 0.02 ≈ 12,980
2. x = (130 - 129.34) / 0.02 ≈ 33.33

Since the number of racquets sold cannot be fractional, we can conclude that the company should manufacture and sell approximately 12,980 racquets to earn a profit of $174,900.