Find the equalibrium price

D=-5p+40
s=-p^2+30p-8

To find the equilibrium price, we need to set the quantity demanded (D) equal to the quantity supplied (S) and solve for the price.

Given:
D = -5p + 40
S = -p^2 + 30p - 8

Setting D equal to S:
-5p + 40 = -p^2 + 30p - 8

Now, let's simplify this equation and solve for p:

Rearrange the equation:
-p^2 + 30p - 8 + 5p - 40 = 0

Combine like terms:
-p^2 + 35p - 48 = 0

This is a quadratic equation in terms of p. To solve it, we can use the quadratic formula:

p = (-b ± √(b^2 - 4ac)) / 2a

For our equation, a = -1, b = 35, and c = -48. Substituting these values into the quadratic formula:

p = (-(35) ± √((35)^2 - 4(-1)(-48))) / 2(-1)

Simplifying further:

p = (-35 ± √(1225 - 192)) / -2

p = (-35 ± √1033) / -2

Now, we have two possible values for p. Let's calculate both:

p1 = (-35 + √1033) / -2
p2 = (-35 - √1033) / -2

p1 ≈ 1.23
p2 ≈ 33.77

Therefore, there are two potential equilibrium prices: p1 ≈ 1.23 and p2 ≈ 33.77.