The demand and supply equations for a certain item are given by
D = -5p + 40
s = -p^2+30p-8
Find the equilibrium price.
Wouldn't you set demand equal to supply, and solve for P? I will be happy to critique your work.
To find the equilibrium price, we need to set the demand equal to the supply.
Given:
Demand equation: D = -5p + 40
Supply equation: S = -p^2 + 30p - 8
Setting demand equal to supply, we have:
-5p + 40 = -p^2 + 30p - 8
Now, let's rearrange this equation to make it a quadratic equation:
0 = -p^2 + 30p - 8 + 5p - 40
0 = -p^2 + 35p - 48
To solve this quadratic equation, we can use factoring, completing the square, or the quadratic formula. Let's use the quadratic formula, which states that for a quadratic equation in the form ax^2 + bx + c = 0, the solutions for x are given by:
x = (-b +/- sqrt(b^2 - 4ac)) / 2a
In our case, a = -1, b = 35, and c = -48.
Using the quadratic formula, we have:
p = (-35 +/- sqrt(35^2 - 4(-1)(-48))) / 2(-1)
Simplifying further, we get:
p = (-35 +/- sqrt(1225 - 192)) / (-2)
p = (-35 +/- sqrt(1033)) / (-2)
Now, evaluating the two possible solutions:
p1 = (-35 + sqrt(1033)) / (-2)
p2 = (-35 - sqrt(1033)) / (-2)
These are the two possible equilibrium prices for the given demand and supply equations.
Note: It's important to double-check the signs and whether it makes logical sense. In some cases, one of the solutions may be extraneous or not practical.