the supply and demand function for a product are qs = (S)p=5p^2-800 and qd= D(p)=2p^2-80p+3500. Determine the market equilibrium price pe and quantity qe

To determine the market equilibrium price (pe) and quantity (qe), we need to set the supply and demand functions equal to each other and solve for the price and quantity at which they intersect.

Supply Function: qs = 5p^2 - 800
Demand Function: qd = 2p^2 - 80p + 3500

Setting the two functions equal to each other:
5p^2 - 800 = 2p^2 - 80p + 3500

Now, let's simplify and solve for p (price):

3p^2 + 80p - 4300 = 0

This is now a quadratic equation, which can be solved by factoring, completing the square, or using the quadratic formula. Let's use the quadratic formula:

p = [-b ± √(b^2 - 4ac)] / (2a)

In our equation, a = 3, b = 80, and c = -4300. Substituting these values into the formula:

p = [-80 ± √(80^2 - 4 * 3 * -4300)] / (2 * 3)

Calculating this expression will give us two possible values for the price, p. Let's denote them as p1 and p2.

After determining the values of p1 and p2, we can substitute the price back into either the supply or demand function to find the corresponding quantity.

For example, let's use the supply function:

qs = 5p^2 - 800

Substituting p1 and p2:

qe1 = 5p1^2 - 800
qe2 = 5p2^2 - 800

So, the market equilibrium price (pe) is the value of p1 or p2 that satisfies both the supply and demand equations, and the market equilibrium quantity (qe) is the corresponding quantity obtained from the supply or demand function.

Note: Sometimes, there may not be a real solution for p, or there may be multiple solutions. In such cases, it is important to assess the problem and consider any additional factors to determine the most appropriate market equilibrium price and quantity.