SOLUTION: D= -200p + 35,000 The supply equation is predicted to be S= -p^2 + 400p - 20,000 Find the equilibrium price.

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons  -> Quadratic Equation Lesson -> SOLUTION: D= -200p + 35,000 The supply equation is predicted to be S= -p^2 + 400p - 20,000 Find the equilibrium price.      Log On


   

Question 33505This question is from textbook
: D= -200p + 35,000
The supply equation is predicted to be
S= -p^2 + 400p - 20,000
Find the equilibrium price. This question is from textbook

Answer by mbarugel(146) About Me  (Show Source):
You can put this solution on YOUR website!
Hello!
This question can be easily solved in the following way.
We call "equilibrium price" to the the price (the value of "p") that makes demand equal supply. So we simply have to set up the equation:
D = S
Since:
D=+-200p+%2B+35000
S=+-p%5E2+%2B+400p+-+20000
Then, from D = S, we get:
-200p+%2B+35000+=+-p%5E2+%2B+400p+-+20000
After combining like terms, we're left with:
-p%5E2+%2B+600p+-+55000+=+0
This quadratic equation can then be solved using the quadratic formula:
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case -1x%5E2%2B600x%2B-55000+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28600%29%5E2-4%2A-1%2A-55000=140000.

Discriminant d=140000 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-600%2B-sqrt%28+140000+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28600%29%2Bsqrt%28+140000+%29%29%2F2%5C-1+=+112.917130661303
x%5B2%5D+=+%28-%28600%29-sqrt%28+140000+%29%29%2F2%5C-1+=+487.082869338697

Quadratic expression -1x%5E2%2B600x%2B-55000 can be factored:
-1x%5E2%2B600x%2B-55000+=+%28x-112.917130661303%29%2A%28x-487.082869338697%29
Again, the answer is: 112.917130661303, 487.082869338697. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+-1%2Ax%5E2%2B600%2Ax%2B-55000+%29



So there are two solutions: p=112.91 or p=487.08. However, notice that the 487.08 solution doesn't make sense, as supply and demand are negative in this case (try plugging p=487.08 into either the supply or demand formula). Therefore, the correct answer is that the equilibrium price is 112.91.

I hope this helps!
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