Question 1199775: Find the equilibrium quantity and the equilibrium price.
p = -2x^2 + 80 and p = 15x + 30
Answer by textot(100)   (Show Source):
You can put this solution on YOUR website! **1. Set Supply Equal to Demand**
* Equilibrium occurs where supply equals demand.
* Set the two equations equal to each other:
-2x^2 + 80 = 15x + 30
**2. Rearrange the Equation**
* 2x^2 + 15x - 50 = 0
**3. Solve for x (Equilibrium Quantity)**
* You can solve this quadratic equation using the quadratic formula:
x = [-b ± √(b² - 4ac)] / 2a
where:
* a = 2
* b = 15
* c = -50
x = [-15 ± √(15² - 4 * 2 * -50)] / (2 * 2)
x = [-15 ± √(625)] / 4
x = [-15 ± 25] / 4
* x1 = (-15 + 25) / 4 = 2.5
* x2 = (-15 - 25) / 4 = -10
Since quantity cannot be negative, we discard x2 = -10.
**Equilibrium Quantity: x = 2.5**
**4. Find the Equilibrium Price**
* Substitute the equilibrium quantity (x = 2.5) into either the supply or demand equation.
* Using the supply equation:
p = 15 * 2.5 + 30
p = 37.5 + 30
p = 67.5
**Equilibrium Price: p = 67.5**
**Therefore, the equilibrium quantity is 2.5 units, and the equilibrium price is 67.5.**
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