SOLUTION: The demand and supply equations for a certain item are given by D=-5p+40 S=-p^2+30p-8 Find the equilibrium price. Please help. Thank you.

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Question 33062: The demand and supply equations for a certain item are given by
D=-5p+40
S=-p^2+30p-8
Find the equilibrium price.
Please help. Thank you.
Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
Equilibrium means D=S
So, -p^2+30p-8=-5p+40
p^2-35p+48=0
a=1, b=-35, c=48
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ap%5E2%2Bbp%2Bc=0 (in our case 1p%5E2%2B-35p%2B48+=+0) has the following solutons:

p%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=%28-35%29%5E2-4%2A1%2A48=1033.

Discriminant d=1033 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--35%2B-sqrt%28+1033+%29%29%2F2%5Ca.

p%5B1%5D+=+%28-%28-35%29%2Bsqrt%28+1033+%29%29%2F2%5C1+=+33.5701586799882
p%5B2%5D+=+%28-%28-35%29-sqrt%28+1033+%29%29%2F2%5C1+=+1.4298413200118

Quadratic expression 1p%5E2%2B-35p%2B48 can be factored:
1p%5E2%2B-35p%2B48+=+1%28p-33.5701586799882%29%2A%28p-1.4298413200118%29
Again, the answer is: 33.5701586799882, 1.4298413200118. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-35%2Ax%2B48+%29

Since item counts must be whole numbers, round
the values of "p" down to a whole number.
Hope this helps.
Cheers,
Stan H.